When rolling a fair six-sided die repeatedly, what is the probability that, before rolling any odd number, you have rolled each even number (2, 4, and 6) at least once?

Free worked solution · Quant Blueprint

1/20 = 5%.

1. Initially a useful even arrives before an odd with probability 3/(3+3) = 1/2.
2. After one even has appeared, ignore repeats of it. Among outcomes that change the state, two new evens compete with three odds, giving probability 2/5 of advancing.
3. After two evens, ignore both repeats. The last new even competes with three odds, giving probability 1/4. Multiply: (1/2)(2/5)(1/4) = 1/20.

Common mistake

Requiring the first three rolls to be different evens is too restrictive. Repeated even faces are allowed before collecting the last one.

Follow-up

With m desired faces and r losing faces, all equally likely, the analogous probability is ∏ₖ₌₁ᵐ k/(k+r) = 1/choose(m+r,m).

Original educational explanation for this practice question. Reviewed 2026-10-13. Firm-style practice; the label describes the collection’s preparation context.