The following problems come from your textbook. For those of you who may not have access to the current edition, I've recreated them below. From your textbook, chapter 13, pages 507 – 509, do the following problems:
13.7
Consider the set of all possible five-card poker hands dealt fairly from a standard deck of fifty-two cards.
a. How many atomic events are there in the joint probability distribution (i.e. how many five-card hands are there)?
b. What is the probability of each atomic event?
c. What is the probability of being dealt a royal straight flush? Four of a kind?
13.15
After your yearly checkup, the doctor has good news and bad news. The bad news is that you tested positive for a serious disease and that the test is 99% accurate (i.e. the probability of testing positive when you do have the test is .99, as is the probability of testing negative when you don't have the disease). The good news is that this is a rare disease, striking only 1 in 10,000 people of your age. Why is it good news that the disease is rare? What are the chances that you actually have the disease?
13.18
Suppose you are given a bag containing n unbiased coins. You are told that n-1 of these coins are normal, with heads on one side and tails on the other, whereas one coin is fake, with heads on both sides.
a. Suppose you reach into the bag, pick out a coin at random, flip it, and get a head. What is the (conditional) probability that the coin you chose is the fake coin?
b. Suppose you continue flipping the coin for a total of k times after picking it and see k heads. Now what is the conditional probability that you picked the fake coin?
c. Suppose you wanted to decide whether the chosen coin was fake by flipping it k times. The decision procedure returns fake if all k flips come up heads; otherwise it returns normal. What is the (unconditional) probability that this procedure makes an error?
13.21
Suppose you are witness to a nighttime hit-and-run accident involving a taxi in Athens. All taxis in Athens are blue or green. You swear, under oath, that the taxi was blue. Extensive testing shows that, under the dim lighting conditions, discrimination between blue and green is 75% reliable.
a. a. Is it possible to calculate the most likely color for the taxi? (Hint: distinguish carefully between the proposition that the taxi is blue and the proposition that it appears blue.)
b. What if you know that 9 out of 10 Athenian taxis are green?
