A drawer holds purple socks and yellow socks. If n socks are taken out of the drawer at random, the probability that all are yellow is 1/2. What is the smallest possible number of socks in the drawer (as a function of n)?

It is equally probable that two signals reach a receiver at any instant of the time T. The receiver will be jammed if the time difference in the reception of the two signals is less than T. Find the probability that the receiver will be jammed.

Jack has just been given a ten-question multiple choice quiz in history class. Each question has five answers, of which only one is correct. Since Jack has not attended class recently, he does not know any of the answers. Assuming Jack guesses randomly on all ten questions, find the probability that he will answer six or more of the questions correctly and avoid an F.

Charlie is about to take two laps in the school swimming pool. The time of his first lap is X minutes,  where X is an Exponential(1) random variable. The time of his second lap is Y minutes, where Y is an Exponential(X) random variable. What is the probability that he completes his second lap within one minute?

Benford's Law claims that numbers chosen from very large data files tend to have "1" as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with "1" as the leading digit is about 0.301. Suppose you are an auditor for a very large corporation. The revenue report involves millions of numbers in a large computer file. Let us say you took a random sample of n = 250 numerical entries from the file and r = 60 of the entries had a first nonzero digit of 1. Let p represent the population proportion of all numbers in the corporate file that have a first nonzero digit of 1. Test the claim that p is less than 0.301 by using α = 0.01. What does the area of the sampling distribution corresponding to your P-value look like?         a. The area in the right tail of the standard normal curve.   b. The area not including the right tail of the standard normal curve.…