Question 1 If we repeatedly roll a balanced die, then, in the long run, it will come up 4 about one-sixth of the time. But what is the probability that such a die will come up "4" exactly once in six rolls? Question 2 According to the Daily Racing Form, the probability is about 0.67 that the favorite in a horse race will finish in the money (first, second, or third place). In the next five races, what is the probability that the favorite finishes in the money a. exactly twice? b. exactly four times? c. at least four times? d. between two and four times, inclusive? e. Determine the probability distribution of the random variable X, the number of times the favorite finishes in the money in the next five races. f. Identify the probability distribution of X as right skewed, symmetric, or left skewed without consulting its probability distribution or drawing its probability histogram. g. Draw a probability histogram for X. h. Obtain the mean and standard deviation of the random variable X. j. Interpret your answer for the mean in words. Question 3 In the paper "The Distribution of Wars in Time (Journal of the Royal Statistical Society, Vol. 107, No. 3/4, pp. 242250), L. F. Richardson analyzed the distribution of wars in time. From the data, we determined that the number of wars that begin during a given calendar year has roughly a Poisson distribution with parameter 1 = 0.7. If a calendar year is selected at random, find the probability that the number, X, of wars that begin during that calendar year will be a. zero. b. at most two. c. between one and three, inclusive. d. Find and interpret the mean of the random variable X. e. Determine the standard deviation of X