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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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