Download Probability Part 4 - Princeton High School

AP Statistics
Probability
Part 3: Probabilities of Compound Events
Part 4: Conditional Probability
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Solve probability problems where events are independent and dependent.
Solve probability problems where events are mutually exclusive &
inclusive.
Solve problems involving conditional probability
Solve problems involving conditional probability where one event is a
subset of the other event.
Probabilities of Compound Events
Probability of Two Independent Events.
 If two events, A and B, are independent, then the probability of
both events occurring is the product of each individual probability.
P(A and B) = P (A) × P(B)
Ex: According to the U.S. Department of Transportation statistics, the top ten
airlines in the United States arrive on time 80% of the time. During vacation, the
Gunther family has direct flights to Pittsburgh, Minneapolis, Denver, and
Nashville on different days. What is the probability that all their flights arrived
on time?
Probability of Two Dependent Events
 If two events, A and B, are dependent, then the probability of both
events occurring is the product of each individual probability.
P(A and B) = P (A) × P(B following A)
Ex: Paul has 3 rock, 4 country, and 2 punk CDs in his car. One day, before he
starts driving, he pulls 2 CDs from his center console without looking. What is
the probability that both CDs are country?
Probability of Mutually Exclusive Events
 Mutually Exclusive – when two events cannot happen at the same
time.
 If two events, A and B, are mutually exclusive,
then the probability that either A or B occurs is
the sum of their probabilities.
P(A or B) = P (A) + P(B)
Ex: Elizabeth is a contestant in a game where if she selects a blue marble or a red
marble she gets an all – expenses paid Alaskan cruise. Elizabeth must select the
marble at random from a bag containing 2 blue, 3 red, 9 yellow, and 10 green
marbles. What is the probability that she will win the cruise?
Probability of Inclusive Events
 Inclusive – when two events can happen at the same time.
 If two events, A and B, are inclusive, then the
probability that either A or B occurs is the sum of
their probabilities decreased by the probability of
both occurring.
P(A or B) = P (A) + P(B) – P(A and B)
Ex: What is the probability of rolling two number cubes, in which the first
number cube shows a 2 or the sum of the number cubes is 6 or 7?
Examples:
1. There are 5 students and 4 teachers on the school publications committee.
A group of 5 members is being selected at random to attend a workshop
on school newspapers. What is the probability that the group attending
the workshop will have at least 3 students?
Conditional Probability
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The probability of an event under the condition that some preceding event
has occurred.
Notation: P (A|B) = the probability of A given B
The conditional probability of event A, given event B, is defined as
P ( A | B) =
P ( AandB)
where P(B) ¹ 0
P(B)
Examples:
2. In a medical research laboratory there is a drug being tested that promotes
hair growth in balding men. The results of a preliminary test are shown in the
table below.
What is the probability that a test subject’s hair grew, given that he used the
experimental drug?
3. A survey is being conducted of families with 3 children. If a family is selected
at random, what is the probability that the family will have exactly 2 boys if the
second child is a boy?
4. Greg tosses two coins. What is the probability that he has tossed 2 heads,
given that he has tossed at least 1 head?
5. A 12 – sided dodecahedron has the numerals 1 through 12 on its faces. The
die is rolled once, and the number on the top face is recorded. What is the
probability that the number is a multiple of 4 if it is known that it is even?