Discrete Math for CS CMPSC 360 LECTURE 34 Last time:

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Discrete Math for CS CMPSC 360 LECTURE 34 Last time: Conditional probability. Bayes’ rule. Today: Total probability rule. CMPSC 360 8/2/2019

Conditional probability: definition Let A and B be events, and suppose that Pr 𝐵 ≠0. The conditional probability Pr 𝐴 𝐵 , the probability of A given B, is Pr 𝐴 𝐵 = Pr 𝐴∩𝐵 Pr⁡[𝐵] . 8/2/2019

Useful conditional probability facts Bayes’ rule. Let A and B be events, and suppose that Pr 𝐵 ≠0. Then Pr 𝐴 𝐵 = Pr 𝐵|𝐴 ⋅ Pr 𝐴 Pr⁡[𝐵] . Total probability rule. Let A and B be events. Then Pr 𝐵 = Pr 𝐵 𝐴 ⋅ Pr 𝐴 + Pr 𝐵 𝐴 ⋅ Pr 𝐴 . 8/2/2019

Clicker question (frequency BC) Test for medical disorder (from last lecture/recitations). We choose a random person from the population. Event A: the chosen person is affected. Event B: the chosen person tests positive. Pr 𝐴 =0.05, Pr 𝐵 𝐴 0.9, Pr 𝐵 𝐴 =0.2. What is the probability of A given that B occurred, Pr⁡[𝐴|𝐵]? < 0.1 In the interval [0.1,0.2). In the interval [0.2,0.3). In the interval [0.3,0.5). ≥0.5 8/2/2019

Tennis match You will play a tennis match against opponent X or Y. If X is chosen, you win with probability 0.7. If Y is chosen, you win with probability 0.3. Your opponent is chosen by flipping a coin with bias 0.6 in favor of X. What is your probability of winning? 8/2/2019

Clicker question (frequency BC) You will play a tennis match against opponent X or Y. If X is chosen, you win with probability 0.7. If Y is chosen, you win with probability 0.3. Your opponent is chosen by flipping a coin with bias 0.6 in favor of X. What is your probability of winning? < 0.3 In the interval [0.3,0.4). In the interval [0.4,0.55). In the interval [0.55,0.7). ≥0.7 8/2/2019

Balls and bins We have two bins with balls. Bin 1 contains 3 black balls and 2 white balls. Bin 2 contains 1 black ball and 1 white ball. We pick a bin uniformly at random. Then we pick a ball uniformly at random from that bin. What is the probability that we picked bin 1, given that we picked a white ball? 8/2/2019