MATH 166 Spring 2016
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1.6 Conditional Probability
Definition: Given two events E and F , we call the probability that E will occur given that F has
occured the conditional probability and write P (E|F ). Read this as “the probability of E given F .”
Let E and F be two events in a sample space S. The conditional probability that E occurs given
that F has occurred is defined to be
P (E|F ) =
P (E ∩ F )
P (F )
Product Rule: If E and F are two events in a sample space S with P (E) > 0 and P (F ) > 0, then
P (E ∩ F ) = P (F )P (E|F ) = P (E)P (F |E)
Examples:
1. In a survey of 1000 eligible voters selected at random, it was found that 100 had a college
degree. Additionally, it was found that 80% of those who had a college degree voted in the last
presidential election, whereas 50% of the people who did not have a college degree voted in the
last presidential election. Assuming that the poll is representative of all eligible voters, find the
probability that an eligible voter selected at random will have the following characteristics.
(a) The voter did not had a college degree and did not vote in the last presidential election.
(b) The voter did not vote in the last presidential election.
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2. A pair of fair 6-sided dice is rolled. What is the probability that a 4 is rolled if it is known that
the sum of the numbers landing uppermost is less than or equal to 6?
3. From the tree diagram find P (A ∩ E), P (A), and P (A|E).
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4. A company surveyed 1000 people on their age and the number of jeans purchased annually.
The results of the poll are shown in the table. A person is selected at random. What is the
probability that the person, who is over 18, purchases 2 pairs of jeans annually?
5. Two machines turn out all the products in a factory, with the first machine producing 75% of
the product and the second 25%. The first machine produces defective products 8% of the time
and the second machine 2% of the time. Draw a tree diagram representing this experiment.
Use this tree diagram to find the probabilities below.
(a) What is the probability that a defective part is produced at this factory given that it was
made on the first machine?
(b) What is the probability that a defective part is produced at this factory?
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6. An experiment consists of two independent trials. The outcomes of the first trial are A, B, and
C, with probabilities of occurring equal to 0.2, 0.4, and 0.4, respectively. The outcomes of the
second trial are E and F , with probabilities of occurring equal to 0.6 and 0.4. Draw a tree
diagram representing this experiment. Use this tree diagram to find the probabilities P (B) and
P (F |B). Does P (B ∩ F ) = P (B)P (F )? Are B and F independent events?
Definition: We say that two events E and F are independent if the outcome of one does not affect
the outcome of the other, that is, if
P (E|F ) = P (E) and P (F |E) = P (F )
Independent Events Theorem: Let E and F be two events with P (E) > 0 and P (F ) > 0. Then
E and F are independent if, and only if,
P (E ∩ F ) = P (E)P (F )
Definition: A set of events {E1 , E2 , . . . , En } is said to be independent if, for any k of these events,
the probability of the intersection of these k events is the product of the probabilities of each of the
k events. This must hold for any k = 2, 3, . . . , n.
Examples:
1. If A and B are independent events, P (A) = 0.25, and P (B) = 0.35, find the probabilities below.
(a) P (A ∩ B)
(b) P (A ∪ B)
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(c) P (A|B)
(d) P (AC ∪ B C )
2. (p. 61) A building has three elevators. The chance that elevator A is not working is 12%, the
chance that elevator B is not working is 15%, and the chance that elevator C is not working is
9%. If these probabilities are independent, what is the probability that exactly one elevator is
not working?
3. The personnel department of Franklin National Life Insurance compiled the accompanying
data regarding the income and education of its employees. Let A be the event that a randomly
chosen employee has a college degree, and let B be the event that the chosen employee’s income
is more than $60, 000. Find P (B|AC ).
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