The product can be odd only if all three factors are odd:

(Thanks, Nick.)

02/01/2021 UPDATE: From reader Paul-Georg Becker:

Another dice quickie:

I have a set of n dice, but unfortunately only one of them is fair. If I roll all dice and *add* the n resulting numbers together, what is the probability that the product will be odd?

Answer:

For natural n let S_{n} be the sum of the resulting numbers x_{1}, …, x_{n}. We assume that x_{n} is the number shown by the fair die. Then we have

P(S_{n} even) = P(x_{n} + S_{n-1} even)

= P(S_{n-1} even) P(x_{n} even) + P(S_{n-1} odd) P(x_{n} odd)

= P(S_{n-1} even) /2 + P(S_{n-1} odd) / 2

= (P(S_{n-1} even) + P(S_{n-1} odd)) / 2 = 1/2

It does not even matter what the result for S_{n-1} is.

(Thanks, Paul-Georg.)