A colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his students, giving a prize of a cookie if they could solve it. Here is one of them.

PROBLEM 116

Bob invites Joe to see his new program, which prints matrices with random integer entries. (These entries are as likely to be even as to be odd.) Joe, being a bit of a gambler, wants to bet Bob a dollar that the next matrix will have an even determinant. Should Bob take the bet? What is the probability that the next matrix will have an even determinant?

Note:
The determinant of the 2 x 2 matrix

is given by .

ADVANCED PROBLEM 116A

What is the probability that the determinant of an matrix with randomly chosen integer entries is divisible by the prime number ?
(Assume that the residues are equally likely for each integer entry.)