A software function generates integers from 1 to n with equal probability. If the function is called three times, what is the probability that the product of the three numbers is even?

Example Answer

Assuming the three calls are independent and n is a positive integer, the product is even unless all three numbers are odd. There are ceiling(n/2) odd numbers from 1 through n, so the probability is 1 minus the cube of ceiling(n/2) divided by n.

When n is even, exactly half the numbers are odd, and the result simplifies to 7/8. When n is odd, the odd numbers are slightly more common, so 7/8 is not exact. For n equal to three, two of the three possible numbers are odd, giving 1 minus (2/3) cubed, or 19/27. For n equal to one, the product is always odd and the probability is zero.

Worked example

P(even product) = 1 − (ceil(n/2)/n)^3, for independent calls and integer n >= 1.

Why this works

The complement calculation handles both parity cases and the smallest valid input. Uniform marginal outputs alone do not establish independence between calls.