MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
An EXPERIMENT is any observation of a random phenomenon.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
An EXPERIMENT is any observation of a random phenomenon.
The set of all possible outcomes for an
experiment is called the sample space.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
An EXPERIMENT is any observation of a random phenomenon.
The set of all possible outcomes for an
experiment is called the sample space.
If you roll a single ordinary die, the set of all possible outcomes is the
sample space for that ‘experiment’.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
An EXPERIMENT is any observation of a random phenomenon.
The set of all possible outcomes for an
experiment is called the sample space.
If you roll a single ordinary die, the set of all possible outcomes is the
sample space for that ‘experiment’.
The sample space is 𝑆 = 1,2,3,4,5,6 .
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
An EXPERIMENT is any observation of a random phenomenon.
The set of all possible outcomes for an
experiment is called the sample space.
If you roll a single ordinary die, the set of all possible outcomes is the
sample space for that ‘experiment’.
The sample space is 𝑆 = 1,2,3,4,5,6 .
If you toss a coin twice and record the result (H or T) for each toss.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
An EXPERIMENT is any observation of a random phenomenon.
The set of all possible outcomes for an
experiment is called the sample space.
If you roll a single ordinary die, the set of all possible outcomes is the
sample space for that ‘experiment’.
The sample space is 𝑆 = 1,2,3,4,5,6 .
If you toss a coin twice and record the result (H or T) for each toss.
The sample space is 𝑆 = 𝐻𝐻, 𝐻𝑇, 𝑇𝐻, 𝑇𝑇 .
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
If you toss a pair of dice and look at the possible rolls, the sample space is:
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
If you toss a pair of dice and look at the possible rolls, the sample space is:
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MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
If you toss a pair of dice and look at the possible rolls, the sample space is:
or, written out as a
set of ordered pairs
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{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(31),(3,2),(3,3),(3,4),(3,5),(3,6),
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
If you spin the spinner on this board 3 times and look at the colors that could
occur (and in what order)…
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
If you spin the spinner on this board 3 times and look at the colors that could
occur (and in what order)…
(by the Fundamental Counting Principle,
the sample space has 3 x 3 x 3 = 27 elements)
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
If you spin the spinner on this board 3 times and look at the colors that could
occur (and in what order)…
(by the Fundamental Counting Principle,
the sample space has 3 x 3 x 3 = 27 elements)
and if you wanted to, you could list all 27…
possibly with the aid of a tree diagram:
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
If you spin the spinner on this board 3 times and look at the colors that could
occur (and in what order)…
(by the Fundamental Counting Principle,
the sample space has 3 x 3 x 3 = 27 elements)
and if you wanted to, you could list all 27…
possibly with the aid of a tree diagram:
{ bbb, bbr, bby, brb, brr, bry, byb, byr, byy, rrr, rrb, rry, rbr, rbb, rby, ryr,
ryb, ryy, yyy, yyb, yyr, yby, ybb, ybr, yry, yrb, yrr }
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
An event is a SUBSET of a SAMPLE SPACE.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
An event is a SUBSET of a SAMPLE SPACE.
Let’s return to the spinner problem.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
An event is a SUBSET of a SAMPLE SPACE.
Let’s return to the spinner problem.
We spin the spinner on this board 3 times and look at the colors that could
occur (and in what order).
{ bbb, bbr, bby, brb, brr, bry, byb, byr, byy, rrr, rrb, rry, rbr,
rbb, rby, ryr, ryb, ryy, yyy, yyb, yyr, yby, ybb, ybr, yry, yrb, yrr }
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
An event is a SUBSET of a SAMPLE SPACE.
Let’s return to the spinner problem.
We spin the spinner on this board 3 times and look at the colors that could
occur (and in what order).
{ bbb, bbr, bby, brb, brr, bry, byb, byr, byy, rrr, rrb, rry, rbr,
rbb, rby, ryr, ryb, ryy, yyy, yyb, yyr, yby, ybb, ybr, yry, yrb, yrr }
Suppose we are interested in the event:
“Red appears exactly twice when the spinner is spun 3 times.”
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
An event is a SUBSET of a SAMPLE SPACE.
Let’s return to the spinner problem.
We spin the spinner on this board 3 times and look at the colors that could
occur (and in what order).
{ bbb, bbr, bby, brb, brr, bry, byb, byr, byy, rrr, rrb, rry, rbr,
rbb, rby, ryr, ryb, ryy, yyy, yyb, yyr, yby, ybb, ybr, yry, yrb, yrr }
Suppose we are interested in the event:
“Red appears exactly twice when the spinner is spun 3 times.”
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
An event is a SUBSET of a SAMPLE SPACE.
Let’s return to the spinner problem.
We spin the spinner on this board 3 times and look at the colors that could
occur (and in what order).
{ bbb, bbr, bby, brb, brr, bry, byb, byr, byy, rrr, rrb, rry, rbr,
rbb, rby, ryr, ryb, ryy, yyy, yyb, yyr, yby, ybb, ybr, yry, yrb, yrr }
Suppose we are interested in the event:
“Red appears exactly twice when the spinner is spun 3 times.”
{ brr, rrb, rry, rbr, ryr, yrr }
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
An event is a SUBSET of a SAMPLE SPACE.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
An event is a SUBSET of a SAMPLE SPACE.
An event with only one possible outcome is called a simple event.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
An event is a SUBSET of a SAMPLE SPACE.
An event with only one possible outcome is called a simple event.
Ex: Rolling a number greater than ‘5’ on the roll of an ordinary die.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
An event is a SUBSET of a SAMPLE SPACE.
An event with only one possible outcome is called a simple event.
Ex: Rolling a number greater than ‘5’ on the roll of an ordinary die.
An event equal to the entire sample space is called a certain event.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
An event is a SUBSET of a SAMPLE SPACE.
An event with only one possible outcome is called a simple event.
Ex: Rolling a number greater than ‘5’ on the roll of an ordinary die.
An event equal to the entire sample space is called a certain event.
Ex: Rolling a number less than ‘7’ on the roll of an ordinary die.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
An event is a SUBSET of a SAMPLE SPACE.
An event with only one possible outcome is called a simple event.
Ex: Rolling a number greater than ‘5’ on the roll of an ordinary die.
An event equal to the entire sample space is called a certain event.
Ex: Rolling a number less than ‘7’ on the roll of an ordinary die.
An event consisting of the empty set is called an impossible event.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
The set of all possible outcomes for an experiment is called the sample space.
An event is a SUBSET of a SAMPLE SPACE.
An event with only one possible outcome is called a simple event.
Ex: Rolling a number greater than ‘5’ on the roll of an ordinary die.
An event equal to the entire sample space is called a certain event.
Ex: Rolling a number less than ‘7’ on the roll of an ordinary die.
An event consisting of the empty set is called an impossible event.
Ex: Rolling a ‘7’ on the roll of an ordinary die.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Because events are sets, we can use set notation and operations with sets.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Because events are sets, we can use set notation and operations with sets.
SET OPERATIONS FOR EVENTS
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Because events are sets, we can use set notation and operations with sets.
SET OPERATIONS FOR EVENTS
Let E and F be events for a sample space S.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Because events are sets, we can use set notation and operations with sets.
SET OPERATIONS FOR EVENTS
Let E and F be events for a sample space S.
𝐸 ∩ 𝐹 means that both events E and F occur.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Because events are sets, we can use set notation and operations with sets.
SET OPERATIONS FOR EVENTS
Let E and F be events for a sample space S.
𝐸 ∩ 𝐹 means that both events E and F occur.
𝐸 ∪ 𝐹 means that either event E or F (or both) occur.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Because events are sets, we can use set notation and operations with sets.
SET OPERATIONS FOR EVENTS
Let E and F be events for a sample space S.
𝐸 ∩ 𝐹 means that both events E and F occur.
𝐸 ∪ 𝐹 means that either event E or F (or both) occur.
𝐸′ means that event 𝐸 does not occur.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Because events are sets, we can use set notation and operations with sets.
Two events that cannot occur at the same time
(such as getting both a head and a tail on the same coin toss)
are called mutually exclusive events.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Because events are sets, we can use set notation and operations with sets.
Two events that cannot occur at the same time
(such as getting both a head and a tail on the same coin toss)
are called mutually exclusive events.
Mathematically, E and F are mutually exclusive events if 𝐸⋂𝐹 = ∅.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Because events are sets, we can use set notation and operations with sets.
Two events that cannot occur at the same time
(such as getting both a head and a tail on the same coin toss)
are called mutually exclusive events.
Mathematically, E and F are mutually exclusive events if 𝐸⋂𝐹 = ∅.
(By definition, mutually exclusive events are disjoint sets.)
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Because events are sets, we can use set notation and operations with sets.
Two events that cannot occur at the same time
(such as getting both a head and a tail on the same coin toss)
are called mutually exclusive events.
Mathematically, E and F are mutually exclusive events if 𝐸⋂𝐹 = ∅.
(By definition, mutually exclusive events are disjoint sets.)
Let S = {1,2,3,4,5,6} be the sample space for rolling a single fair die.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Because events are sets, we can use set notation and operations with sets.
Two events that cannot occur at the same time
(such as getting both a head and a tail on the same coin toss)
are called mutually exclusive events.
Mathematically, E and F are mutually exclusive events if 𝐸⋂𝐹 = ∅.
(By definition, mutually exclusive events are disjoint sets.)
Let S = {1,2,3,4,5,6} be the sample space for rolling a single fair die.
Suppose O = {1,3,5} and E = {2,4,6} are the events that a die roll is
even or odd, respectively.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Because events are sets, we can use set notation and operations with sets.
Two events that cannot occur at the same time
(such as getting both a head and a tail on the same coin toss)
are called mutually exclusive events.
Mathematically, E and F are mutually exclusive events if 𝐸⋂𝐹 = ∅.
(By definition, mutually exclusive events are disjoint sets.)
Let S = {1,2,3,4,5,6} be the sample space for rolling a single fair die.
Suppose O = {1,3,5} and E = {2,4,6} are the events that a die roll is
even or odd, respectively.
O and E are mutually exclusive events because O⋂E = ∅.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Now we are ready to begin our study of probability.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Now we are ready to begin our study of probability.
The probability of an outcome in a sample space, S, is a
number between 0 and 1, inclusive satisfying these conditions:
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Now we are ready to begin our study of probability.
The probability of an outcome in a sample space, S, is a
number between 0 and 1, inclusive satisfying these conditions:
 The sum of the probabilities of all possible outcomes is 1.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Now we are ready to begin our study of probability.
The probability of an outcome in a sample space, S, is a
number between 0 and 1, inclusive satisfying these conditions:
 The sum of the probabilities of all possible outcomes is 1.
 The probability of an event 𝐸, 𝑃(𝐸), is the sum of the
probabilities of the outcomes that make up 𝐸.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Now we are ready to begin our study of probability.
The probability of an outcome in a sample space, S, is a
number between 0 and 1, inclusive satisfying these conditions:
 The sum of the probabilities of all possible outcomes is 1.
 The probability of an event 𝐸, 𝑃(𝐸), is the sum of the
probabilities of the outcomes that make up 𝐸.
Putting all of this together we get these
BASIC PROPERTIES OF PROBABILITY
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Now we are ready to begin our study of probability.
The probability of an outcome in a sample space, S, is a
number between 0 and 1, inclusive satisfying these conditions:
 The sum of the probabilities of all possible outcomes is 1.
 The probability of an event 𝐸, 𝑃(𝐸), is the sum of the
probabilities of the outcomes that make up 𝐸.
Putting all of this together we get these
BASIC PROPERTIES OF PROBABILITY
1. 0 ≤ 𝑃(𝐸) ≤ 1
2. 𝑃 ∅ = 0
3. 𝑃 𝑆 = 1
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Now we are ready to begin our study of probability.
The probability of an outcome in a sample space, S, is a
number between 0 and 1, inclusive satisfying these conditions:
 The sum of the probabilities of all possible outcomes is 1.
 The probability of an event 𝐸, 𝑃(𝐸), is the sum of the
probabilities of the outcomes that make up 𝐸.
Putting all of this together we get these
BASIC PROPERTIES OF PROBABILITY
1. 0 ≤ 𝑃(𝐸) ≤ 1
2. 𝑃 ∅ = 0
3. 𝑃 𝑆 = 1
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
Now we are ready to begin our study of probability.
The probability of an outcome in a sample space, S, is a
number between 0 and 1, inclusive satisfying these conditions:
 The sum of the probabilities of all possible outcomes is 1.
 The probability of an event 𝐸, 𝑃(𝐸), is the sum of the
probabilities of the outcomes that make up 𝐸.
Putting all of this together we get these
BASIC PROPERTIES OF PROBABILITY
1. 0 ≤ 𝑃(𝐸) ≤ 1
2. 𝑃 ∅ = 0
3. 𝑃 𝑆 = 1
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
EMPIRICAL ASSIGNMENT OF PROBABILITIES (relative frequency of event)
If we perform an experiment multiple times and 𝐸 is a possible event in
the experiment, then we can estimate the probability of
𝐸 the number of times E occurs follows:
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
EMPIRICAL ASSIGNMENT OF PROBABILITIES (relative frequency of event)
If we perform an experiment multiple times and 𝐸 is a possible event in
the experiment, then we can estimate the probability of
𝐸 the number of times E occurs follows:
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑃 𝐸 =
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 𝑖𝑠 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑒𝑑
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
EMPIRICAL ASSIGNMENT OF PROBABILITIES (relative frequency of event)
If we perform an experiment multiple times and 𝐸 is a possible event in
the experiment, then we can estimate the probability of
𝐸 the number of times E occurs follows:
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑃 𝐸 =
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 𝑖𝑠 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑒𝑑
A drug company is testing a new flu vaccine. A patient is injected with the
vaccine and observed for side effects. This is done 100 times with these results:
Side Effects
None
Mild
Severe
Number of times
72
25
3
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
EMPIRICAL ASSIGNMENT OF PROBABILITIES (relative frequency of event)
If we perform an experiment multiple times and 𝐸 is a possible event in
the experiment, then we can estimate the probability of
𝐸 the number of times E occurs follows:
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑃 𝐸 =
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 𝑖𝑠 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑒𝑑
A drug company is testing a new flu vaccine. A patient is injected with the
vaccine and observed for side effects. This is done 100 times with these results:
Side Effects
None
Mild
Severe
Number of times
72
25
3
3
P severe =
= 0.03
100
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
EMPIRICAL ASSIGNMENT OF PROBABILITIES (relative frequency of event)
If we perform an experiment multiple times and 𝐸 is a possible event in
the experiment, then we can estimate the probability of
𝐸 the number of times E occurs follows:
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑃 𝐸 =
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 𝑖𝑠 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑒𝑑
A drug company is testing a new flu vaccine. A patient is injected with the
vaccine and observed for side effects. This is done 100 times with these results:
Side Effects
None
Mild
Severe
Number of times
72
25
3
3
P severe =
= 0.03
100
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
EMPIRICAL ASSIGNMENT OF PROBABILITIES (relative frequency of event)
If we perform an experiment multiple times and 𝐸 is a possible event in
the experiment, then we can estimate the probability of
𝐸 the number of times E occurs follows:
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑃 𝐸 =
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 𝑖𝑠 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑒𝑑
A drug company is testing a new flu vaccine. A patient is injected with the
vaccine and observed for side effects. This is done 100 times with these results:
Side Effects
None
Mild
Severe
Number of times
72
25
3
3
P severe =
= 0.03
100
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
EMPIRICAL ASSIGNMENT OF PROBABILITIES (relative frequency of event)
If we perform an experiment multiple times and 𝐸 is a possible event in
the experiment, then we can estimate the probability of
𝐸 the number of times E occurs follows:
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑃 𝐸 =
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 𝑖𝑠 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑒𝑑
A drug company is testing a new flu vaccine. A patient is injected with the
vaccine and observed for side effects. This is done 100 times with these results:
Side Effects
None
Mild
Severe
Number of times
72
25
3
3
P severe =
= 0.03
100
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
EMPIRICAL ASSIGNMENT OF PROBABILITIES (relative frequency of event)
If we perform an experiment multiple times and 𝐸 is a possible event in
the experiment, then we can estimate the probability of
𝐸 the number of times E occurs follows:
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑃 𝐸 =
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 𝑖𝑠 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑒𝑑
A drug company is testing a new flu vaccine. A patient is injected with the
vaccine and observed for side effects. This is done 100 times with these results:
Side Effects
None
Mild
Severe
Number of times
72
25
3
3
P severe =
= 0.03
100
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
If we want to calculate the probability of an event E from a sample
space with equally likely outcomes, we can use the following result:
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
If we want to calculate the probability of an event E from a sample
space with equally likely outcomes, we can use the following result:
BASIC PROBABILITY PRINCIPLE (BPP)
If 𝑆 is an equally-likely sample space and event 𝐸 is a subset of 𝑆, then
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
If we want to calculate the probability of an event E from a sample
space with equally likely outcomes, we can use the following result:
BASIC PROBABILITY PRINCIPLE (BPP)
If 𝑆 is an equally-likely sample space and event 𝐸 is a subset of 𝑆, then
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
If we want to calculate the probability of an event E from a sample
space with equally likely outcomes, we can use the following result:
BASIC PROBABILITY PRINCIPLE (BPP)
If 𝑆 is an equally-likely sample space and event 𝐸 is a subset of 𝑆, then
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
If we want to calculate the probability of an event E from a sample
space with equally likely outcomes, we can use the following result:
BASIC PROBABILITY PRINCIPLE (BPP)
If 𝑆 is an equally-likely sample space and event 𝐸 is a subset of 𝑆, then
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
Word of Warning: Although this is an extremely powerful result and one that
we will use over and over again, keep in mind that it only applies when the
elements of the sample space are equally likely to occur.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
If we want to calculate the probability of an event E from a sample
space with equally likely outcomes, we can use the following result:
BASIC PROBABILITY PRINCIPLE (BPP)
If 𝑆 is an equally-likely sample space and event 𝐸 is a subset of 𝑆, then
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
Word of Warning: Although this is an extremely powerful result and one that
we will use over and over again, keep in mind that it only applies when the
elements of the sample space are equally likely to occur.
For example, if we flip a fair coin and want to calculate 𝑃 𝐻𝑒𝑎𝑑 , then
we can use the BPP because heads and tails are equally likely.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
If we want to calculate the probability of an event E from a sample
space with equally likely outcomes, we can use the following result:
BASIC PROBABILITY PRINCIPLE (BPP)
If 𝑆 is an equally-likely sample space and event 𝐸 is a subset of 𝑆, then
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
Word of Warning: Although this is an extremely powerful result and one that
we will use over and over again, keep in mind that it only applies when the
elements of the sample space are equally likely to occur.
If we flip a coin biased to land more often on heads, then we can’t use
the BPP to find 𝑃 𝐻𝑒𝑎𝑑 because heads and tails are not equally likely.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
BASIC PROBABILITY PRINCIPLE (BPP)
If 𝑆 is an equally-likely sample space and event 𝐸 is a subset of 𝑆, then
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
Alice and John agree to flip a fair coin to settle a dispute. What is the
probability that the coin comes up heads?
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
BASIC PROBABILITY PRINCIPLE (BPP)
If 𝑆 is an equally-likely sample space and event 𝐸 is a subset of 𝑆, then
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
Alice and John agree to flip a fair coin to settle a dispute. What is the
probability that the coin comes up heads?
Sample Space: { H , T }
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
BASIC PROBABILITY PRINCIPLE (BPP)
If 𝑆 is an equally-likely sample space and event 𝐸 is a subset of 𝑆, then
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
Alice and John agree to flip a fair coin to settle a dispute. What is the
probability that the coin comes up heads?
Sample Space: { H , T }
By the BPP above, 𝑃 𝐻𝑒𝑎𝑑 =
# 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑔𝑒𝑡 𝑎 ℎ𝑒𝑎𝑑
# 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒
=
1
2
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
BASIC PROBABILITY PRINCIPLE (BPP)
If 𝑆 is an equally-likely sample space and event 𝐸 is a subset of 𝑆, then
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
Alice and John agree to flip a fair coin to settle a dispute. What is the
probability that the coin comes up heads?
Sample Space: { H , T }
By the BPP above, 𝑃 𝐻𝑒𝑎𝑑 =
# 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑔𝑒𝑡 𝑎 ℎ𝑒𝑎𝑑
# 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒
=
1
2
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
BASIC PROBABILITY PRINCIPLE (BPP)
If 𝑆 is an equally-likely sample space and event 𝐸 is a subset of 𝑆, then
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
Alice and John agree to flip a fair coin to settle a dispute. What is the
probability that the coin comes up heads?
Sample Space: { H , T }
By the BPP above, 𝑃 𝐻𝑒𝑎𝑑 =
# 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑔𝑒𝑡 𝑎 ℎ𝑒𝑎𝑑
# 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒
=
1
2
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
BASIC PROBABILITY PRINCIPLE (BPP)
If 𝑆 is an equally-likely sample space and event 𝐸 is a subset of 𝑆, then
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
Alice and John agree to flip a fair coin to settle a dispute. What is the
probability that the coin comes up heads?
Sample Space: { H , T }
By the BPP above, 𝑃 𝐻𝑒𝑎𝑑 =
# 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑔𝑒𝑡 𝑎 ℎ𝑒𝑎𝑑
# 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒
=
1
2
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
BASIC PROBABILITY PRINCIPLE (BPP)
If 𝑆 is an equally-likely sample space and event 𝐸 is a subset of 𝑆, then
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
Alice and John agree to flip a fair coin to settle a dispute. What is the
probability that the coin comes up heads?
Sample Space: { H , T }
By the BPP above, 𝑃 𝐻𝑒𝑎𝑑 =
# 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑔𝑒𝑡 𝑎 ℎ𝑒𝑎𝑑
# 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒
=
1
2
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
BASIC PROBABILITY PRINCIPLE (BPP)
If 𝑆 is an equally-likely sample space and event 𝐸 is a subset of 𝑆, then
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
Alice and John agree to flip a fair coin to settle a dispute. What is the
probability that the coin comes up heads?
Sample Space: { H , T }
By the BPP above, 𝑃 𝐻𝑒𝑎𝑑 =
# 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑔𝑒𝑡 𝑎 ℎ𝑒𝑎𝑑
# 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒
=
1
2
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
BASIC PROBABILITY PRINCIPLE (BPP)
If 𝑆 is an equally-likely sample space and event 𝐸 is a subset of 𝑆, then
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
Alice and John agree to flip a fair coin to settle a dispute. What is the
probability that the coin comes up heads?
Sample Space: { H , T }
By the BPP above, 𝑃 𝐻𝑒𝑎𝑑 =
# 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑔𝑒𝑡 𝑎 ℎ𝑒𝑎𝑑
# 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒
=
1
2
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
BASIC PROBABILITY PRINCIPLE (BPP)
If 𝑆 is an equally-likely sample space and event 𝐸 is a subset of 𝑆, then
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
Suppose you roll 2 dice. What is the probability that the sum is 9?
RecallMATH
that this
is the
space for
a 2 dice
roll problem.
110
Sec sample
13-1 Lecture:
Intro
to Probability
11
12
Intro13 to Probability
14
15
16
BASIC
PRINCIPLE
22 PROBABILITY
23
24
25 (BPP)26
If 𝑆 is an equally-likely
and event
subset of 𝑆, then
31
32 sample
33 space 34
35 𝐸 is a36
41 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟
42
43
44 𝑒𝑣𝑒𝑛𝑡45
46
𝑜𝑓 𝑡𝑖𝑚𝑒𝑠
𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 = 51
=
52
53
54
55
56
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
21
61
62
63
64
65
66
Suppose you roll 2 dice. What is the probability that the sum is 9?
RecallMATH
that this
is the
space for
a 2 dice
roll problem.
110
Sec sample
13-1 Lecture:
Intro
to Probability
11
12
Intro13 to Probability
14
15
16
BASIC
PRINCIPLE
22 PROBABILITY
23
24
25 (BPP)26
If 𝑆 is an equally-likely
and event
subset of 𝑆, then
31
32 sample
33 space 34
35 𝐸 is a36
41 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟
42
43
44 𝑒𝑣𝑒𝑛𝑡45
46
𝑜𝑓 𝑡𝑖𝑚𝑒𝑠
𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 = 51
=
52
53
54
55
56
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
21
61
62
63
64
65
66
Suppose you roll 2 dice. What is the probability that the sum is 9?
𝑛 𝑆 = 36
RecallMATH
that this
is the
space for
a 2 dice
roll problem.
110
Sec sample
13-1 Lecture:
Intro
to Probability
11
12
Intro13 to Probability
14
15
16
BASIC
PRINCIPLE
22 PROBABILITY
23
24
25 (BPP)26
If 𝑆 is an equally-likely
and event
subset of 𝑆, then
31
32 sample
33 space 34
35 𝐸 is a36
41 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟
42
43
44 𝑒𝑣𝑒𝑛𝑡45
46
𝑜𝑓 𝑡𝑖𝑚𝑒𝑠
𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 = 51
=
52
53
54
55
56
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
21
61
62
63
64
65
66
Suppose you roll 2 dice. What is the probability that the sum is 9?
𝑛 𝑆 = 36
RecallMATH
that this
is the
space for
a 2 dice
roll problem.
110
Sec sample
13-1 Lecture:
Intro
to Probability
11
12
Intro13 to Probability
14
15
16
BASIC
PRINCIPLE
22 PROBABILITY
23
24
25 (BPP)26
If 𝑆 is an equally-likely
and event
subset of 𝑆, then
31
32 sample
33 space 34
35 𝐸 is a36
41 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟
42
43
44 𝑒𝑣𝑒𝑛𝑡45
46
𝑜𝑓 𝑡𝑖𝑚𝑒𝑠
𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 = 51
=
52
53
54
55
56
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
21
61
62
63
64
65
66
Suppose you roll 2 dice. What is the probability that the sum is 9?
𝑛 𝐸 =4
𝑛 𝑆 = 36
RecallMATH
that this
is the
space for
a 2 dice
roll problem.
110
Sec sample
13-1 Lecture:
Intro
to Probability
11
12
Intro13 to Probability
14
15
16
BASIC
PRINCIPLE
22 PROBABILITY
23
24
25 (BPP)26
If 𝑆 is an equally-likely
and event
subset of 𝑆, then
31
32 sample
33 space 34
35 𝐸 is a36
41 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟
42
43
44 𝑒𝑣𝑒𝑛𝑡45
46
𝑜𝑓 𝑡𝑖𝑚𝑒𝑠
𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 = 51
=
52
53
54
55
56
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
21
61
62
63
64
65
66
Suppose you roll 2 dice. What is the probability that the sum is 9?
𝑛(𝑠𝑢𝑚 𝑖𝑠 9)
4
1
𝑃 𝑠𝑢𝑚 𝑖𝑠 9 =
=
=
𝑛(𝑆)
36 9
𝑛 𝐸 =4
𝑛 𝑆 = 36
RecallMATH
that this
is the
space for
a 2 dice
roll problem.
110
Sec sample
13-1 Lecture:
Intro
to Probability
11
12
Intro13 to Probability
14
15
16
BASIC
PRINCIPLE
22 PROBABILITY
23
24
25 (BPP)26
If 𝑆 is an equally-likely
and event
subset of 𝑆, then
31
32 sample
33 space 34
35 𝐸 is a36
41 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟
42
43
44 𝑒𝑣𝑒𝑛𝑡45
46
𝑜𝑓 𝑡𝑖𝑚𝑒𝑠
𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 = 51
=
52
53
54
55
56
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
21
61
62
63
64
65
66
Suppose you roll 2 dice. What is the probability that the sum is 9?
𝑛(𝑠𝑢𝑚 𝑖𝑠 9)
4
1
𝑃 𝑠𝑢𝑚 𝑖𝑠 9 =
=
=
𝑛(𝑆)
36 9
𝑛 𝐸 =4
𝑛 𝑆 = 36
Recall that
this110
is the
space for
a 2to
dice
roll problem.
MATH
Secsample
13-1 Lecture:
Intro
Probability
11
12
Intro13 to Probability
14
15
16
BASIC
PRINCIPLE
22 PROBABILITY
23
24
25 (BPP)26
If 𝑆 is an equally-likely
and event
subset of 𝑆, then
31
32 sample
33 space 34
35 𝐸 is a36
41 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟
42
43
44 𝑒𝑣𝑒𝑛𝑡45
46
𝑜𝑓 𝑡𝑖𝑚𝑒𝑠
𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛(𝐸)
𝑃 𝐸 = 51
=
52
53
54
55
56
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑛(𝑆)
21
61
62
63
64
65
66
Suppose you roll 2 dice. What is the probability that the sum is 9?
𝑛(𝑠𝑢𝑚 𝑖𝑠 9)
4
1
𝑃 𝑠𝑢𝑚 𝑖𝑠 9 =
=
=
𝑛(𝑆)
36 9
𝑛 𝐸 =4
𝑛 𝑆 = 36
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
We often use the term odds to express probabilities.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
We often use the term odds to express probabilities.
An online story published on October 20, 2014, referencing the
Vegas Superbook odds board, reportedly had Alabama and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds of 5 to 1 against (often written 5 : 1 against).
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
We often use the term odds to express probabilities.
An online story published on October 20, 2014, referencing the
Vegas Superbook odds board, reportedly had Alabama and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds of 5 to 1 against (often written 5 : 1 against).
But what does that mean?
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
We often use the term odds to express probabilities.
An online story published on October 20, 2014, referencing the
Vegas Superbook odds board, reportedly had Alabama and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds of 5 to 1 against (often written 5 : 1 against).
One way of thinking
about it is to think of
the numbers as
being associated
with slips of paper.
But what does that mean?
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
We often use the term odds to express probabilities.
An online story published on October 20, 2014, referencing the
Vegas Superbook odds board, reportedly had Alabama and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds of 5 to 1 against (often written 5 : 1 against).
One way of thinking
about it is to think of
the numbers as
being associated
with slips of paper.
But what does that mean?
NOT NOT NOT NOT NOT
Win
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
We often use the term odds to express probabilities.
An online story published on October 20, 2014, referencing the
Vegas Superbook odds board, reportedly had Alabama and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds of 5 to 1 against (often written 5 : 1 against).
One way of thinking
about it is to think of
the numbers as
being associated
with slips of paper.
But what does that mean?
NOT NOT NOT NOT NOT
Win
MATH 110 Sec 13-1 Lecture: Intro to Probability
Also, when someone says that for an
Intro to Probability
event the odds are ‘5 to 1 against’,
that also means that
the odds are
ODDS
‘1 to
5 INthe
FAVOR
‘ that to
event.
We often
use
termofodds
express probabilities.
An online story published on October 20, 2014, referencing the
Vegas Superbook odds board, reportedly had Alabama and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds of 5 to 1 against (often written 5 : 1 against).
One way of thinking
about it is to think of
the numbers as
being associated
with slips of paper.
But what does that mean?
NOT NOT NOT NOT NOT
Win
MATH 110 Sec 13-1 Lecture: Intro to Probability
Also, when someone says that for an
Intro to Probability
event the odds are ‘5 to 1 against’,
that also means that
the odds are
ODDS
‘1 to
5 INthe
FAVOR
‘ that to
event.
We often
use
termofodds
express probabilities.
An online story published on October 20, 2014, referencing the
Vegas Superbook odds board, reportedly had Alabama and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds of 5 to 1 against (often written 5 : 1 against).
One way of thinking
about it is to think of
the numbers as
being associated
with slips of paper.
But what does that mean?
NOT NOT NOT NOT NOT
Win
We could also say the odds are 1 : 5 in favor.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Let’s take a moment to be sure that we
Intro
to
Probability
correctly understand the notation.
ODDS
IfWe
we often
say that
thethe
odds
of anodds
eventtoare
‘a : b against’,
then a is
use
term
express
probabilities.
thestory
‘against’
numberon
but
if we say20,
that
the odds
are
An online
published
October
2014,
referencing
the
‘a : bSuperbook
in favor of ‘odds
that event,
a is the ‘in
of’ number.
Vegas
board,then
reportedly
hadfavor
Alabama
and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds of 5 to 1 against (often written 5 : 1 against).
One way of thinking
about it is to think of
the numbers as
being associated
with slips of paper.
But what does that mean?
NOT NOT NOT NOT NOT
Win
We could also say the odds are 1 : 5 in favor.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Let’s take a moment to be sure that we
Intro
to
Probability
correctly understand the notation.
ODDS
IfWe
we often
say that
thethe
odds
of anodds
eventtoare
‘a : b against’,
then a is
use
term
express
probabilities.
thestory
‘against’
numberon
but
if we say20,
that
the odds
are
An online
published
October
2014,
referencing
the
‘a : bSuperbook
in favor of ‘odds
that event,
a is the ‘in
of’ number.
Vegas
board,then
reportedly
hadfavor
Alabama
and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds of 5 to 1 against (often written 5 : 1 against).
One way of thinking
about it is to think of
the numbers as
being associated
with slips of paper.
But what does that mean?
NOT NOT NOT NOT NOT
Win
We could also say the odds are 1 : 5 in favor.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Let’s take a moment to be sure that we
Intro
to
Probability
correctly understand the notation.
ODDS
IfWe
we often
say that
thethe
odds
of anodds
eventtoare
‘a : b against’,
then a is
use
term
express
probabilities.
thestory
‘against’
numberon
but
if we say20,
that
the odds
are
An online
published
October
2014,
referencing
the
‘a : bSuperbook
in favor of ‘odds
that event,
a is the ‘in
of’ number.
Vegas
board,then
reportedly
hadfavor
Alabama
and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds of 5 to 1 against (often written 5 : 1 against).
One way of thinking
about it is to think of
the numbers as
being associated
with slips of paper.
But what does that mean?
NOT NOT NOT NOT NOT
Win
We could also say the odds are 1 : 5 in favor.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Let’s take a moment to be sure that we
Intro
to
Probability
correctly understand the notation.
ODDS
IfWe
we often
say that
thethe
odds
of anodds
eventtoare
‘a : b against’,
then a is
use
term
express
probabilities.
thestory
‘against’
numberon
but
if we say20,
that
the odds
are
An online
published
October
2014,
referencing
the
‘a : bSuperbook
in favor of ‘odds
that event,
a is the ‘in
of’ number.
Vegas
board,then
reportedly
hadfavor
Alabama
and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds
to 1 against
(often
So, in of
the5example
at hand
‘5 : 1written
against’5 : 1 against).
One way of thinking
about it is to think of
the numbers as
being associated
with slips of paper.
But what does that mean?
NOT NOT NOT NOT NOT
Win
We could also say the odds are 1 : 5 in favor.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Let’s take a moment to be sure that we
Intro
to
Probability
correctly understand the notation.
ODDS
IfWe
we often
say that
thethe
odds
of anodds
eventtoare
‘a : b against’,
then a is
use
term
express
probabilities.
thestory
‘against’
numberon
but
if we say20,
that
the odds
are
An online
published
October
2014,
referencing
the
‘a : bSuperbook
in favor of ‘odds
that event,
a is the ‘in
of’ number.
Vegas
board,then
reportedly
hadfavor
Alabama
and Ole
in favor college
of
Miss tied for the team most likelyagainst
to win the 2015
football
playoff with odds
to 1 against
(often
So, in of
the5example
at hand
‘5 : 1written
against’5 : 1 against).
One way of thinking
about it is to think of
the numbers as
being associated
with slips of paper.
But what does that mean?
NOT NOT NOT NOT NOT
Win
We could also say the odds are 1 : 5 in favor.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Let’s take a moment to be sure that we
Intro
to
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correctly understand the notation.
ODDS
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We could also say the odds are 1 : 5 in favor.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Let’s take a moment to be sure that we
Intro
to
Probability
correctly understand the notation.
ODDS
IfWe
we often
say that
thethe
odds
of anodds
eventtoare
‘a : b against’,
then a is
use
term
express
probabilities.
thestory
‘against’
numberon
but
if we say20,
that
the odds
are
An online
published
October
2014,
referencing
the
‘a : bSuperbook
in favor of ‘odds
that event,
a is the ‘in
of’ number.
Vegas
board,then
reportedly
hadfavor
Alabama
and Ole
in favor college
of
Miss tied for the team most likelyagainst
to win the 2015
football
playoff with odds
to 1 against
(often
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the5example
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‘5 : 1written
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mean?
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while ‘1 NOT
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the numbers as
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MATH 110 Sec 13-1 Lecture: Intro to Probability
Also, when someone says that for an
Intro to Probability
event the odds are ‘5 to 1 against’,
that also means that
the odds are
ODDS
‘1 to
5 INthe
FAVOR
‘ that to
event.
We often
use
termofodds
express probabilities.
An online story published on October 20, 2014, referencing the
Vegas Superbook odds board, reportedly had Alabama and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds of 5 to 1 against (often written 5 : 1 against).
One way of thinking
about it is to think of
the numbers as
being associated
with slips of paper.
But what does that mean?
NOT NOT NOT NOT NOT
Win
We could also say the odds are 1 : 5 in favor.
MATH 110 Sec 13-1 Lecture: Intro to Probability
But if the odds against Alabama
Intro to Probability
winning the 2015 college football
playoff are
5 : 1, what is the
ODDS
probability
of Alabama
We often use
the term
odds to winning
expressit?
probabilities.
An online story published on October 20, 2014, referencing the
Vegas Superbook odds board, reportedly had Alabama and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds of 5 to 1 against (often written 5 : 1 against).
One way of thinking
about it is to think of
the numbers as
being associated
with slips of paper.
NOT NOT NOT NOT NOT
Win
MATH 110 Sec 13-1 Lecture: Intro to Probability
But if the odds against Alabama
Intro to Probability
winning the 2015 college football
playoff are
5 : 1, what is the
ODDS
probability
of Alabama
We often use
the term
odds to winning
expressit?
probabilities.
An online story published on October 20, 2014, referencing the
Vegas Superbook odds board, reportedly had Alabama and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds of 5 to 1 against (often written 5 : 1 against).
One way of thinking
about it is to think of
the numbers as
being associated
with slips of paper.
NOT NOT NOT NOT NOT
𝑛(𝑊𝑖𝑛) 1
By the BPP: 𝑃 𝑈𝐴 𝑤𝑖𝑛𝑠 𝑖𝑡 =
=
𝑛(𝑆)
6
Win
MATH 110 Sec 13-1 Lecture: Intro to Probability
But if the odds against Alabama
Intro to Probability
winning the 2015 college football
playoff are
5 : 1, what is the
ODDS
probability
of Alabama
We often use
the term
odds to winning
expressit?
probabilities.
An online story published on October 20, 2014, referencing the
Vegas Superbook odds board, reportedly had Alabama and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds of 5 to 1 against (often written 5 : 1 against).
One way of thinking
about it is to think of
the numbers as
being associated
with slips of paper.
NOT NOT NOT NOT NOT
𝑛(𝑊𝑖𝑛) 1
By the BPP: 𝑃 𝑈𝐴 𝑤𝑖𝑛𝑠 𝑖𝑡 =
=
𝑛(𝑆)
6
Win
MATH 110 Sec 13-1 Lecture: Intro to Probability
But if the odds against Alabama
Intro to Probability
winning the 2015 college football
playoff are
5 : 1, what is the
ODDS
probability
of Alabama
We often use
the term
odds to winning
expressit?
probabilities.
An online story published on October 20, 2014, referencing the
Vegas Superbook odds board, reportedly had Alabama and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds of 5 to 1 against (often written 5 : 1 against).
One way of thinking
about it is to think of
the numbers as
being associated
with slips of paper.
NOT NOT NOT NOT NOT
𝑛(𝑊𝑖𝑛) 1
By the BPP: 𝑃 𝑈𝐴 𝑤𝑖𝑛𝑠 𝑖𝑡 =
=
𝑛(𝑆)
6
Win
MATH 110 Sec 13-1 Lecture: Intro to Probability
But if the odds against Alabama
Intro to Probability
winning the 2015 college football
playoff are
5 : 1, what is the
ODDS
probability
of Alabama
We often use
the term
odds to winning
expressit?
probabilities.
An online story published on October 20, 2014, referencing the
Vegas Superbook odds board, reportedly had Alabama and Ole
Miss tied for the team most likely to win the 2015 college football
playoff with odds of 5 to 1 against (often written 5 : 1 against).
One way of thinking
about it is to think of
the numbers as
being associated
with slips of paper.
NOT NOT NOT NOT NOT
𝑛(𝑊𝑖𝑛) 1
By the BPP: 𝑃 𝑈𝐴 𝑤𝑖𝑛𝑠 𝑖𝑡 =
=
𝑛(𝑆)
6
Win
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
We often use the term odds to express probabilities.
The previous discussion should make it clear just how closely
probabilities and odds are related.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
We often use the term odds to express probabilities.
The previous discussion should make it clear just how closely
probabilities and odds are related.
If the odds against an event E are a : b
(which is the same as saying the odds in favor are b : a),
𝑏
then 𝑃 𝐸 =
.
𝑎+𝑏
MATH 110 Sec 13-1 Lecture: Intro to Probability
The key thing is this:
Intro to Probability
No matter which way the odds statement is written
(against or in favor), the
probability is always the ‘in
ODDS
favor’ use
number
sumto
of express
the two numbers.
We often
the over
termthe
odds
probabilities.
The previous discussion should make it clear just how closely
probabilities and odds are related.
If the odds against an event E are a : b
(which is the same as saying the odds in favor are b : a),
𝑏
then 𝑃 𝐸 =
.
𝑎+𝑏
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
You may also be given
the probability of an event
Weand
often
usewhat
thethe
term
odds
to express
probabilities.
asked
odds
against
(or in favor
of) are.
If you look
closely atshould
the formula
youjust
will how
see closely
The previous
discussion
makebelow,
it clear
just now
easy
thatare
is. related.
probabilities
and
odds
If the odds against an event E are a : b
(which is the same as saying the odds in favor are b : a),
𝑏
then 𝑃 𝐸 =
.
𝑎+𝑏
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
You may also be given
the probability of an event
Weand
often
usewhat
thethe
term
odds
to express
probabilities.
asked
odds
against
(or in favor
of) are.
If you look
closely atshould
the formula
youjust
will how
see closely
The previous
discussion
makebelow,
it clear
just now
easy
thatare
is. related.
probabilities
and
odds
If the odds against an event E are a : b
(which is the same as saying the odds in favor are b : a),
b is the ‘in favor of’
𝑏
then 𝑃 𝐸 =
. number and once you
𝑎+𝑏
realize that, it is easy
to find a.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
You may also be given
the probability of an event
Weand
often
usewhat
thethe
term
odds
to express
probabilities.
asked
odds
against
(or in favor
of) are.
If you look
closely atshould
the formula
youjust
will how
see closely
The previous
discussion
makebelow,
it clear
just now
easy
thatare
is. related.
probabilities
and
odds
If the odds against an event E are a : b
(which is the same as saying the odds in favor are b : a),
b is the ‘in favor of’
𝑏
then 𝑃 𝐸 =
. number and once you
𝑎+𝑏
1
6
We know that 𝑃 𝑈𝐴 𝑤𝑖𝑛𝑠 𝑖𝑡 = =
1
5+1
realize that, it is easy
to find a.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
You may also be given
the probability of an event
Weand
often
usewhat
thethe
term
odds
to express
probabilities.
asked
odds
against
(or in favor
of) are.
If you look
closely atshould
the formula
youjust
will how
see closely
The previous
discussion
makebelow,
it clear
just now
easy
thatare
is. related.
probabilities
and
odds
If the odds against an event E are a : b
(which is the same as saying the odds in favor are b : a),
b is the ‘in favor of’
𝑏
then 𝑃 𝐸 =
. number and once you
𝑎+𝑏
1
6
We know that 𝑃 𝑈𝐴 𝑤𝑖𝑛𝑠 𝑖𝑡 = =
1
5+1
realize that, it is easy
to find a.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
You may also be given
the probability of an event
Weand
often
usewhat
thethe
term
odds
to express
probabilities.
asked
odds
against
(or in favor
of) are.
If you look
closely atshould
the formula
youjust
will how
see closely
The previous
discussion
makebelow,
it clear
just now
easy
thatare
is. related.
probabilities
and
odds
If the odds against an event E are a : b
(which is the same as saying the odds in favor are b : a),
b is the ‘in favor of’
𝑏
then 𝑃 𝐸 =
. number and once you
𝑎+𝑏
1
6
We know that 𝑃 𝑈𝐴 𝑤𝑖𝑛𝑠 𝑖𝑡 = =
1
5+1
b
realize that, it is easy
to find a.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
You may also be given
the probability of an event
Weand
often
usewhat
thethe
term
odds
to express
probabilities.
asked
odds
against
(or in favor
of) are.
If you look
closely atshould
the formula
youjust
will how
see closely
The previous
discussion
makebelow,
it clear
just now
easy
thatare
is. related.
probabilities
and
odds
If the odds against an event E are a : b
(which is the same as saying the odds in favor are b : a),
b is the ‘in favor of’
𝑏
then 𝑃 𝐸 =
. number and once you
a
1
We know that 𝑃 𝑈𝐴 𝑤𝑖𝑛𝑠 𝑖𝑡 = =
6
𝑎+𝑏
1
5+1
b
realize that, it is easy
to find a.
MATH
Secagainst
13-1 Lecture:
Introwinning
to Probability
So the110
odds
Alabama
the
Introfootball
to Probability
2015 college
playoff are 5 : 1.
ODDS
You may also be given
the probability of an event
Weand
often
usewhat
thethe
term
odds
to express
probabilities.
asked
odds
against
(or in favor
of) are.
If you look
closely atshould
the formula
youjust
will how
see closely
The previous
discussion
makebelow,
it clear
just now
easy
thatare
is. related.
probabilities
and
odds
If the odds against an event E are a : b
(which is the same as saying the odds in favor are b : a),
b is the ‘in favor of’
𝑏
then 𝑃 𝐸 =
. number and once you
a
1
We know that 𝑃 𝑈𝐴 𝑤𝑖𝑛𝑠 𝑖𝑡 = =
6
𝑎+𝑏
1
5+1
b
realize that, it is easy
to find a.
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
Finally, it is also possible to relate probability and odds with this result:
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
Finally, it is also possible to relate probability and odds with this result:
If 𝑃(𝐸) is the probability of event 𝐸,
(or
𝑃(𝐸 ′ )
The odds against 𝐸 is
𝑃(𝐸)
more commonly written 𝑃 𝐸 ′ ∶
𝑃 𝐸 .)
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
Finally, it is also possible to relate probability and odds with this result:
If 𝑃(𝐸) is the probability of event 𝐸,
(or
𝑃(𝐸 ′ )
The odds against 𝐸 is
𝑃(𝐸)
more commonly written 𝑃 𝐸 ′ ∶
𝑃 𝐸 .)
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
Finally, it is also possible to relate probability and odds with this result:
If 𝑃(𝐸) is the probability of event 𝐸,
(or
𝑃(𝐸 ′ )
The odds against 𝐸 is
𝑃(𝐸)
more commonly written 𝑃 𝐸 ′ ∶
𝑃 𝐸 .)
For example, if 𝑃 𝐸 =0.3, then 𝑃 𝐸 ′ = 1 − 𝑃 𝐸 = 1 − 0.3 = 0.7
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
Finally, it is also possible to relate probability and odds with this result:
If 𝑃(𝐸) is the probability of event 𝐸,
(or
𝑃(𝐸 ′ )
The odds against 𝐸 is
𝑃(𝐸)
more commonly written 𝑃 𝐸 ′ ∶
𝑃 𝐸 .)
For example, if 𝑃 𝐸 =0.3, then 𝑃 𝐸 ′ = 1 − 𝑃 𝐸 = 1 − 0.3 = 0.7
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
Finally, it is also possible to relate probability and odds with this result:
If 𝑃(𝐸) is the probability of event 𝐸,
(or
𝑃(𝐸 ′ )
The odds against 𝐸 is
𝑃(𝐸)
more commonly written 𝑃 𝐸 ′ ∶
𝑃 𝐸 .)
For example, if 𝑃 𝐸 =0.3, then 𝑃 𝐸 ′ = 1 − 𝑃 𝐸 = 1 − 0.3 = 0.7
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
Finally, it is also possible to relate probability and odds with this result:
If 𝑃(𝐸) is the probability of event 𝐸,
(or
𝑃(𝐸 ′ )
The odds against 𝐸 is
𝑃(𝐸)
more commonly written 𝑃 𝐸 ′ ∶
𝑃 𝐸 .)
For example, if 𝑃 𝐸 =0.3, then 𝑃 𝐸 ′ = 1 − 𝑃 𝐸 = 1 − 0.3 = 0.7
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
Finally, it is also possible to relate probability and odds with this result:
If 𝑃(𝐸) is the probability of event 𝐸,
(or
𝑃(𝐸 ′ )
The odds against 𝐸 is
𝑃(𝐸)
more commonly written 𝑃 𝐸 ′ ∶
𝑃 𝐸 .)
For example, if 𝑃 𝐸 =0.3, then 𝑃 𝐸 ′ = 1 − 𝑃 𝐸 = 1 − 0.3 = 0.7
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
Finally, it is also possible to relate probability and odds with this result:
If 𝑃(𝐸) is the probability of event 𝐸,
(or
𝑃(𝐸 ′ )
The odds against 𝐸 is
𝑃(𝐸)
more commonly written 𝑃 𝐸 ′ ∶
𝑃 𝐸 .)
For example, if 𝑃 𝐸 =0.3, then 𝑃 𝐸 ′ = 1 − 𝑃 𝐸 = 1 − 0.3 = 0.7
So, the odds against 𝐸 is
𝑃(𝐸 ′ )
0.7
7
=
=
𝑃(𝐸)
0.3
3
or 7 : 3 .
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
Finally, it is also possible to relate probability and odds with this result:
If 𝑃(𝐸) is the probability of event 𝐸,
(or
𝑃(𝐸 ′ )
The odds against 𝐸 is
𝑃(𝐸)
more commonly written 𝑃 𝐸 ′ ∶
𝑃 𝐸 .)
For example, if 𝑃 𝐸 =0.3, then 𝑃 𝐸 ′ = 1 − 𝑃 𝐸 = 1 − 0.3 = 0.7
So, the odds against 𝐸 is
𝑃(𝐸 ′ )
0.7
7
=
=
𝑃(𝐸)
0.3
3
or 7 : 3 .
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
Finally, it is also possible to relate probability and odds with this result:
If 𝑃(𝐸) is the probability of event 𝐸,
(or
𝑃(𝐸 ′ )
The odds against 𝐸 is
𝑃(𝐸)
more commonly written 𝑃 𝐸 ′ ∶
𝑃 𝐸 .)
For example, if 𝑃 𝐸 =0.3, then 𝑃 𝐸 ′ = 1 − 𝑃 𝐸 = 1 − 0.3 = 0.7
So, the odds against 𝐸 is
𝑃(𝐸 ′ )
0.7
7
=
=
𝑃(𝐸)
0.3
3
or 7 : 3 .
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
Finally, it is also possible to relate probability and odds with this result:
If 𝑃(𝐸) is the probability of event 𝐸,
(or
𝑃(𝐸 ′ )
The odds against 𝐸 is
𝑃(𝐸)
more commonly written 𝑃 𝐸 ′ ∶
𝑃 𝐸 .)
For example, if 𝑃 𝐸 =0.3, then 𝑃 𝐸 ′ = 1 − 𝑃 𝐸 = 1 − 0.3 = 0.7
So, the odds against 𝐸 is
𝑃(𝐸 ′ )
0.7
7
=
=
𝑃(𝐸)
0.3
3
or 7 : 3 .
MATH 110 Sec 13-1 Lecture: Intro to Probability
Intro to Probability
ODDS
Finally, it is also possible to relate probability and odds with this result:
If 𝑃(𝐸) is the probability of event 𝐸,
(or
𝑃(𝐸 ′ )
The odds against 𝐸 is
𝑃(𝐸)
more commonly written 𝑃 𝐸 ′ ∶
𝑃 𝐸 .)
For example, if 𝑃 𝐸 =0.3, then 𝑃 𝐸 ′ = 1 − 𝑃 𝐸 = 1 − 0.3 = 0.7
So, the odds against 𝐸 is
𝑃(𝐸 ′ )
0.7
7
=
=
𝑃(𝐸)
0.3
3
or 7 : 3 .
You will seldom, if ever, have to convert to odds using this formula unless you want to.
Standard deck of 52 cards
You must familiarize yourself with a standard deck of 52 playing cards.
Standard deck of 52 cards
26 black cards
26 red cards
Standard deck of 52 cards
Standard deck of 52 cards
4 suits (CLUBS, SPADES, HEARTS, DIAMONDS)
13 CLUBS
13 SPADES
13 HEARTS
13 DIAMONDS
Standard deck of 52 cards
4 suits (CLUBS, SPADES, HEARTS, DIAMONDS)
12 Face Cards
Standard deck of 52 cards
4 suits (CLUBS, SPADES, HEARTS, DIAMONDS)
4
Aces
4
4
4
4
Twos Threes Fours Fives
4
4
4
4
Sixes Sevens Eights Nines
4
Tens
4
4
4
Jacks Queens Kings