Quickie - Futility Closet

Quickie

If I roll three dice and multiply the three resulting numbers together, what is the probability that the product will be odd?

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The product can be odd only if all three factors are odd: $\displaystyle \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$ (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 x₁, …, 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.)