Rain and Wind
The local weather channel just released a statistic for the months of November and December. It said that the probability that it would rain on a windy day is 0.3 and the probability that it would rain on a non-windy day is 0.8. The probability of a day being windy is 0.2. As a student in EECS 70, you are curious to play around with these numbers. Find the probability that:
(a) A given day is both windy and rainy.
(b) A given day is rainy.
(c) For a given pair of days, exactly one of the two days is rainy. (You may assume that the weather on the first day does not affect the weather on the second.)
2 Lie Detector
A lie detector is known to be 4/5 reliable when the person is guilty and 9/10 reliable when the person is innocent. If a suspect is chosen from a group of suspects of which only 1/100 have ever committed a crime, and the test indicates that the person is guilty, what is the probability that he is innocent?
3 Bag of Coins
Your friend Forest has a bag of n coins. You know that k are biased with probability p (i.e. these coins have probability p of being heads). Let F be the event that Forest picks a fair coin, and let B be the event that Forest picks a biased coin. Forest draws three coins from the bag, but he does not know which are biased and which are fair.
(a) What is the probability of three coins being pulled in the order FFB?
(b) What is the probability that the third coin he draws is biased?
(c) What is the probability of picking at least two fair coins?
(d) Given that Forest flips the second coin and sees heads, what is the probability that this coin is biased?
