Given two fair dice, what is the probability that both show six if it is observed that at least one die shows a six?

Free worked solution · Quant Blueprint

1/11.

1. The condition leaves the six pairs (6,1),…,(6,6) and the five additional pairs (1,6),…,(5,6).
2. All eleven are equally likely because the original 36 ordered outcomes were equally likely. Only (6,6) has two sixes.
3. Equivalently P(both six | at least one six) = (1/36)/(11/36) = 1/11.

Common mistake

If a specified die is known to show six, the answer is 1/6. That is a different observation from at least one die showing six.

Follow-up

For an independent face with probability p on each die, P(both | at least one) = p²/(2p−p²) = p/(2−p).

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