MATH 141-501
Section 7.2 Lecture Notes
Experiments and Probability
Recall:
• Experiment
• Sample Space = set of all possible observed results
• Outcome = element of the sample space
• Event = subset of the sample space
A probability is a number associated to an outcome in a sample space. The
probability of an event tells how likely an event is to occur.
Symbol: P (x) - probability of outcome x.
Rules of Probability
1.
2.
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Probability Distribution:
A probability distribution is a table which gives the probability of each
outcome of a sample space.
Example 2 (Rolling a Fair Die): The probability distribution for a fair
six-sided die is given below.
Outcome
1
2
3
4
5
6
Probability
1/6
1/6
1/6
1/6
1/6
1/6
This is an example of a
outcome has the same probability.
sample space – one where each
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Determining Probability Theoretically
Example 3: A toin is cossed three times (experiment) and the result of
each toss is recorded (outcome).
What is the sample space?
What is the probability...
... that all three tosses are tails?
... that the first toss is a head, and the next two are tails?
Is this an example of a uniform sample space?
This probability has been determined theoretically, i.e.
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Determining Probability Experimentally
Often, an experiment is repeated many times.
The number of times each outcome occurs is recorded.
The relative frequency of an outcome is given by the fraction
number of times outcome occurred
.
number of times experiment occurred
The relative frequency can be used as a way of assigning probability to outcomes of an experiment.
Example 4: The grade distribution for a certain class is given by the table
below.
A
12
B
20
C
15
D
8
F
8
What is the relative frequency of the outcome “A” for this experiment?
If we select a random student in this class, what is the probability that that
person got an “A”?
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Theoretical Probability vs. Experimental Probability
Example: John Kerrich was a mathematician taken prisoner of war by the
Germans during World War II.
He tossed a coin 10,000 times and recorded whether each toss landed heads or
tails.
After 10,000 tosses, he had recorded 5,067 heads.
Question 1: Based on Kerrich’s experiment, what is the experimental
probability of getting heads when flipping a coin?
Question 2: Does this support or challenge the theoretical probability of
heads having probability 12 ?
(Source: http://yunwah.wordpress.com/2009/10/29/the-patron-saint-of-experimentalscientists/)
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Probability of Events
We can also assign probabilities to events, not just outcomes. For this, we
use the following definition.
If E is an event in a sample space which contains outcomes x1 , x2 , . . . , xn ,
then
P (E) = P (x1 ) + P (x2 ) + . . . + P (xn ).
Procedure: To get the probability of an event,
1. Find the outcomes in that event.
2. Find the probabilities of each outcome in that event.
3.
Example: If a coin is flipped three times, what is the probability that exactly two of the flips are heads?
Note: If S is a uniform sample space, the probability of an event E is
n(E)
n(S)
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Example: If a fair, six-sided die is rolled and the number facing up is
recorded, what is the probability that the number facing up is . . .
. . . greater than 3?
. . . even?
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Example - Heavy Dice: Heavy Dice are a special kind of weighted dice
used by gamblers who are trying to cheat. A casino had its employees roll
one Heavy die a million times. Based on that experiment, the casino’s Senior
Mathematician developed the following probability distribution.
Number on Die
Probability
1
1
24
2
1
3
3
1
24
4
1
12
5
1
4
6
1
4
If a gambler is using a Heavy die, what is the probability that she rolls...
... a number greater than 4?
... an even number?
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Section 7.3
Properties and Rules of Probability
Basic Properties of Probability
If S is the sample space of an experiment, and E is an event of the experiment...
... what do we know about P (E)?
... what do we know about P (S)?
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Example: An experiment consists of rolling two dice. What is the probability of the event that either die shows a 6?
Setup: Let E be the event that the first die shows a 6, and F be the event
that the second die shows a 6. We want to find P (E ∪ F ).
Probability Property 3(a.): If E and F are events in a probability space
S, then
P (E ∪ F ) =
.
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Background - Standard Deck of Cards
A standard deck of cards contains 52 cards.
Each card has a suit and a face value.
There are 4 suits: diamonds (♦), hearts (♥), clubs(♣), and spades(♠).
There are 13 face values: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King.
A card is specified by giving its face value and suit.
Ace of Hearts = A♥
2 of Spades = 2♠
etc.
Diamonds and hearts are red cards.
Clubs and spades are black cards.
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Example
Drawing Cards at Random A card is drawn at random from a standard
deck of cards. What is the probability that the card is a queen or a heart?
Setup: Let Q be the event that a queen is drawn, and H be the event that a
heart is drawn. The probability that we are looking for is
Formula:
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Disjoint Events
Example: A gambling game is played as follows: the player rolls two dice.
If the sum of the numbers two dice is equal to 7 or 11, the player wins. Otherwise, the player loses.
For a particular roll, what is the probability that the player wins?
Setup: Let E = event that player rolls a 7, and F = event that player rolls
an 11.
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How would we describe the event that the player wins this game?
Probability Property 3(b.):If E and F are disjoint events in a probability
space S, then
P (E ∪ F ) =
.
Apply this formula to solve this question.
Solution:
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Complement Rule
Probability Property 4 (Complement Rule): If E is an event, then
probability of the event E c is
P (E c ) = 1 − P (E).
Example: If a card is drawn at random, what is the probability that it is
not a queen?
Example: If a card is drawn at random, what is the probability that it is
not a queen AND not a heart?
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Example:
Two cards are drawn from a deck at random. What is the probability that...
...both cards are queens?
...at least one of the cards is a king or a queen?
...there are no aces among the cards?
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Summary – Properties of Probability
Here is a list of the properties of probability discussed in this section.
Probability Property 1: If E is an event in a sample space S, then
0 ≤ P (E) ≤ 1.
Probability Property 2: If S is a sample space, P (S) = 1.
Probability Propert 3(a.): If E and F are any events in a probability
space S, then
P (E ∪ F ) = P (E) + P (F ) − P (E ∩ F ).
Probability Property 3(b.):If E and F are disjoint events in a probability
space S, then
P (E ∪ F ) = P (E) + P (F ).
Probability Property 4 (Complement Rule): If E is an event, then
probability of the event E c is
P (E c ) = 1 − P (E).
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