Problem 3.61 R. Wolf (1882) threw a die 20, 000 times and recorded the number of times each of the six faces appeared. The results follow. Face 123456 Frequency 3407 3631 3176 2916 3448 3422 (Source: D. J. Hand, et. al. (1994), Small Data Sets, Chapman& Hall, London.)

(a) Compute an upper bound on the probability that the number of times the number five appeared is greater than or equal to the observed value 3448. Assume that the dice are fair and use Chebyshev’s inequality to obtain the upper bound.

(b) Compute an upper bound on the probability that the number of times the number four appeared is less than or equal to the observed value 2916.