Problem of the Week, #116

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 2 \times 2 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

A =  \left(  \begin{array}{cc}  a & b \\  c & d \\  \end{array}  \right)

is given by det(A) = ad - bc.

ADVANCED PROBLEM 116A

What is the probability that the determinant of an n \times n matrix with randomly chosen integer entries is divisible by the prime number p?
(Assume that the residues 0, 1, 2, ... , p-1 are equally likely for each integer entry.)

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