Probability
Q: What is the probability that Steelers win the Super Bowl Sunday?
10%, 25%, 37% All arrived by personal opinion.
My estimate is 86% since they have won 6 of the 7 in which they played. Mine is based on
relative frequency i.e. the number of times they have won (6) to the number of times they played
(7). The other probabilities are based on opinion, also called subjective probabilities. A final
type of probability is classical. This is when the there is a set of finite outcomes and the
probability is the chance a certain subset of that finite set occurs. E.g. flipping a coin and getting
a Head (1 out of 2) or rolling a die and getting a six (1 out of 6) or drawing a Heart from a deck of
cards (13 out of 52 or 1/4).
Sampling with and without replacement.
Which of the following (or both) are true?
1. Law of large numbers: Over a long run period the expected mean or proportion
of the population would be experienced. E.g. flip a coin 1000 times; we would
expect about 500 heads and tails.
2. Law of small numbers: If a pattern occurs. E.g. 6 tails in a row then we are more
likely to see a head on the next flip. Not True!
Probability is the chance that something is the case or that an event will occur. Range of
probabilities is 0 to 1.
Trial is one number of repetitions in an experiment/study.
Sample Space is the set all possible outcomes. E.g. coin flip->head, tail; die roll-> 1,2,3,4,5,6
Event is some outcome or subset of Sample Space: E.g. get a tail on a coin flip; E.g. get an
even number when die rolled.
Complement is a set of outcomes not in the event of interest. E.g. if event of interest is “get a tail
on coin flip” a complement is “not a tail” I.E. get a head. E.g. If event is get an even number on a
roll of die” I.E. we get a 2 or 4 or 6. Complement would be 1, 3, or 5. If the event is “pass the
class with C or better” then the complement is not pass the class meaning get a D or F….NOTE:
getting just a D or getting just an F are not the complement but only part of the complement. The
complement consists of ALL outcomes NOT in the event of interest.
Intersection is when two or more events overlap. i.e. share common outcomes. E.g. let event E
be get even number on die roll (2,4,6), Let S be event we get a 6.
Union: the event of all outcomes of two or more events. E.g. Let E be even numbers on role of
die, let F be get a 5, the union of events E and F would be (2,4,5,6)
Conditional Probability is the probability that event occurs given that another event has already
happened. E.g. Let E be event get an even number on roll of die. Say we know the outcome is an
even number, what is probability we got a 6? 1/3
Independent trials one event does not influence the other event. E.g. coin flip.
Mutually Exclusive of Disjoint Events: events that do not have any common outcomes. Both
events cannot occur at the same time. E.g. let E be even number on roll of die, Let F be we get a
5.
1
Simple Events: are events of only one outcome. E.g. getting a 5.
Notation:
If we define two events as A, and B then
P(A): is probability that event A occurs
P(AC ) or P(A1): the complement of A
P(AorB) or P(AUB: The union of A and B]
A stat class has 4 teaching assistants.
3 females( Lauren, Rona, Leila)
1 male (Tom)
Each Ta is on chare of one section, Let event W be the Ta is female and let event M be TA is
male.
1) Which event is a simple event? Event M
2) Are the events W and M mutually exclusive? Yes Since the events do not overlap. I.E.
share any common outcomes.
3) If two students, unknown to each other, randomly select a TA are their choices
independent? Yes
4) What is probability that a student randomly selects Lauren? ¼
5) You are told your section TA is female. What is probability that Lauren is your TA?
1/3
Mathematical Rules:
- 0 <= P(event) <= 1
- P(event) + P(event complement) = 1
- P(A U B) = P(A) + P(B) - P(A and B)
- If two events, say A and B, are independent then P(A and B) = P(A)*P(B)
- Conditional Probability: P(A|B) means “probability event A occurs given event B has
already occurred.” P(A|B) = P(A and B)/P(B)
- Alternatively, P(B|A) = P(A and B)/P(A)
Note then that if the two events are independent then P(A|B) = P(A) and P(B|A) = P(B) For
example, what is the probability that on two draws from a card deck that both cards are Aces?
That is, find probability that the first card is Ace AND second card is an Ace? Well probability
of first card is Ace is 4/52 or 1/13. Then for both to be an ace we can find the conditional
probability of second card is Ace given first card was and Ace. This conditional probability is
3/51. So P(first and second are Aces) equals the 1/13*3/51 or 3/663
Terminology
Probability: The likelihood that some outcome occurs. The range of any probability is from 0 to
1. The most common calculation of this value comes from the proportion of times that the
outcome would occur in a long run of observations.
Trial: One of a number of repetitions of an experiment.
Independent Trials: Trials are considered independent if the outcome of any one trial is not
affected/influenced by the outcome of any other trial.
2
Sample Space: The set of all possible outcomes. The sum of all individual probabilities within a
sample space is one.
Event: A subset of the sample space.
Complement: The complement of an event consists of all outcomes in the sample space that are
not in that event.
Intersection: The intersection of two events consists of all outcomes that in both events.
Pictorially, this means that the two events overlap.
Union: The union of two events consists of all outcomes that are either in the one event and/or in
the other event.
Conditional Probability: Conditional probability is the probability that an event occurs given
that another event has already occurred.
Mutually Exclusive or Disjoint: Two events are considered mutually exclusive, or disjoint, if
they do not share any common outcomes. Pictorially, this means that the two events do not
overlap.
Notation
If we define two events as A and B then;
P(A): is the probability that event A occurs.
P(Ac): is the complement to A
P(A or B) of P(A U B): The union of events A and B
P(A and B) of P(A ∩ B): The intersection of events A and B
P(A|B): The conditional probability of event A occurring given that event B has occurred. The
vertical slash “|” represents given. The formula for finding conditional probability is:
P(A|B)= P(A and B)/P(B) [read the “probability of A and B, divided by probability of B.”]
Independence and Union of Two Events
Two events, say event A and event B, are independent if any one of the following can be proven
true.
P(A and B) = P(A)*P(B)
P(A|B) = P(A)
P(B|A) = P(B)
The union of these two events is found as follows:
P(A or B) = P(A) + P(B) – P(A and B). [note that if A and B are mutually exclusive the P(A and
B) is zero.
3
Scenarios
Illustrate how common sense applies:
Example 1: With a die ask what is probability of getting a 6? Answer 1/6 and demonstrate this by
letting event A = “getting a 6” and P(A) = 1/6
Example 2: With the die what is probability of getting an even number? Answer 3/6 or 1/2 where
we let event B = “getting and even number” and P(B) = 1/2
Example 3: Say I roll the die and tell you that the number is even (i.e. given you this event
occurred ---- conditioned on this information) what is the probability that the number is a 6?
Answer, with the given information of being even we are down to 3 possible outcomes: 2, 4, or 6
and from this only one outcome is the event of interest a 6 making the probability of getting a 6
when knowing the number is even is 1/3. Using the notation and formula, we let event A =
getting a 6 and let B = getting an even number. From our conditional formula we have P(A|B) =
P(A and B)/P(B) = (1/6)/1/2) = 2/6 = 1/3
FOR EXAM
Assume the distribution of blood types, among Americans is approximately as follows and that
blood types of married couples are independent:
40% type A
25% type B
30% type O
5% type AB
Question: What is the probability that one person of a randomly chosen couple has type A blood
and the other has type O? (Answer: 0.24)
Either the wife can have O and husband A, or wife has A and husband O. Since blood types are
stated as independent P(A and O) is P(A)*P(O) = 0.4 * 0.3 = 0.12 and since P(O and A) is also
possible and also equals 0.12 the final answer is 0.12 + 0.12 = 0.24
Question: An individual with type B blood can safely receive transfusions only from persons with
type B or type O blood. What is the probability that the wife of a man with type B blood is an
acceptable blood donor for him? (Answer: 0.475)
Want to find P(B or O) = P(B) + P(O) - P(B and O). = 0.25 + 0.30 - 0.25*0.30 = .55 - 0.075 =
0.475
Question: You finally graduated from college and are interviewing for two jobs. You estimate the
probability of receiving a job offer from Company A to be 0.50 and the probability is 0.30 that
both Company A and Company B will offer you a job. Given Company A offers you a job, what
is the probability that Company B will also offer you a job? (Answer: 0.6)
With given statement question is asking to find P(B|A) = P(A and B)/P(A) = 0.3/0.5 = 0.6
4