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Announcements Finite Probability Wednesday, October 12th I MyMathLab 5 is due Monday Oct 17 I Problem Set 5 is due Wednesday Oct 19 Today: Sec. 6.4: Conditional Probability Explain the intuitive meaning of conditional probability Calculate conditional probabilities using the definition, including equally likely outcomes Next Class: Sec. 6.4: Conditional Probability II Cherveny Oct 12 Math 1004: Probability Rolling a Die Example Roll a fair die. If you know that the outcome is an odd number, what is the probability that it is a 3? Answer: There are three odd outcomes, {1, 3, 5}. They are equally likely. Since we are told the roll was one of them, the probability of a 3 given the knowledge that the roll is odd is 31 . Cherveny Oct 12 Math 1004: Probability Conditional Probability Definition The conditional probability of “event E given event F ” is P(E |F ) = P(E ∩ F ) P(F ) Formally: When event F is known to have happened, we think of F as a new sample space. New events are subsets of F , which are old events (subsets of the old sample space) intersected with F . And we divide by P(F ) so that the new sample space has probability 1. Cherveny Oct 12 Math 1004: Probability Rolling a Die Example Roll a fair die. If you know that the outcome is an odd number, what is the probability that it is a 3? Answer: P(3|odd) = P(3 ∩ odd) = P(odd) Outcome 1 3 5 Cherveny Oct 12 1 6 1 2 = 1 3 Probability 1/3 1/3 1/3 Math 1004: Probability Conditional Probability Example Example A pair of fair dice is rolled. What is the probability that the sum of the dice is 8, given that exactly one of the dice shows a 3? Answer: P(sum is 8 and exactly one 3) P(exactly one 3) 2/36 1 = = 10/36 5 P(sum is 8|exactly one 3) = Note: When outcomes are equally likely we still solve by counting: P(E |F ) = Cherveny # outcomes in both E and F # outcomes in F Oct 12 Math 1004: Probability Math Club Example The math club has six sophomore and five freshmen members. If three members are selected at random for a competition, what is the probability they are all sophomores, given that at least one is a sophomore? Answer: # teams all sophomore # teams ≥ 1 sophomore C (6, 3) = C (11, 3) − C (5, 3) P(all sophomore| ≥ 1 sophomore) = Cherveny Oct 12 Math 1004: Probability Practice 1. Let E and F be events in sample space S. Suppose P(E ) = 1/2, P(F ) = 1/2, and P(E ∪ F ) = 7/12. Calculate (a) P(E ∩ F ) (b) P(E |F ) (c) P(F |E ) 2. Two cards are drawn one after another from a standard 52 card deck without replacement. What is the probability.. (a) (b) (c) (d) (e) the second is red given the first is red? the second is a heart given the first is a club? the second is the ace of clubs given the first is the ace of clubs? the first card is red given the second card is red? the second card is a heart given the first card is a heart if you replace the first card after you draw it? 3. What is the probability that the sum of two dice is 9 given that exactly one of the dice is a 4? Cherveny Oct 12 Math 1004: Probability Practice Answers 1. Let E and F be events in sample space S. Suppose P(E ) = 1/4, P(F ) = 1/2, and P(E ∪ F ) = 7/12. Calculate (a) P(E ∩ F ) = 1/6 (b) P(E |F ) = 2/3 (c) P(F |E ) = 1/3 2. Two cards are drawn one after another from a standard 52 card deck without replacement. What is the probability.. (a) (b) (c) (d) (e) 25/51 13/51 0 25/51 1/4 3. P(sum 9|exactly one 4) = 1/5 Cherveny Oct 12 Math 1004: Probability