Description

CS 70 Discrete Mathematics and Probability Theory

1 Box of Marbles
You are given two boxes: one of them containing 900 red marbles and 100 blue marbles, the other one contains 500 red marbles and 500 blue marbles.
(a) If we pick one of the boxes randomly, and pick a marble what is the probability that it is blue?
(b) If we see that the marble is blue, what is the probability that it is chosen from box 1?
(c) Suppose we pick one marble from box 1 and without looking at its color we put it aside. Then we pick another marble from box 1. What is the probability that the second marble is blue?
2 Duelling Meteorologists
Tom is a meteorologist in New York. On days when it snows, Tom correctly predicts the snow
70% of the time. When it doesn’t snow, he correctly predicts no snow 95% of the time. In New York, it snows on 10% of all days.
(a) If Tom says that it is going to snow, what is the probability it will actually snow?
(b) Let A be the event that, on a given day, Tom predicts the weather correctly. What is P(A)?
(c) Tom’s friend Jerry is a meteorologist in Alaska. Jerry claims that she is a better meteorologist than Tom even though her overall accuracy is lower. After looking at their records, you determine that Jerry is indeed better than Tom at predicting snow on snowy days and sun on sunny day. Give an instance of the situation described above. Hint: what is the weather like in Alaska?
3 Binary Conditional Probabilities
Let us consider a sample space Ω= {ω1,…,ωN} of size N > 2, and two probability functions P1 and P2 on it. That is, we have two probability spaces: (Ω,P1) and (Ω,P2).
If for every subset A ⊂ Ω of size |A| = 2 and every outcome ω ∈ Ω it is true that P1 (ω | A)= P2 (ω | A), then is it necessarily true that P1 (ω)=P2 (ω) for all ω ∈ Ω? That is, if P1 and P2 are equal conditional on events of size 2, are they equal unconditionally? (Hint: Remember that probabilities must add up to 1.)