Conditional Probability Questions Built on Two Dice

Conditioning on at least one six versus a specific die showing six changes the answer from one eleventh to one sixth, a common source of error.

Two fair six sided dice are rolled, and three standard conditioning questions come up again and again in interviews. Given at least one die shows a six, the probability both show six is one eleventh. Given the sum is eight, the probability at least one die shows a three is two fifths. Given at least one die is even, the probability both are even is one third. Each answer comes from the same mechanical step, count the outcomes consistent with the condition, then count how many of those also satisfy the target event.

A pair of fair dice has 36 equally likely outcomes when both faces are treated as distinguishable, one through six on each die. Conditional probability restricts attention to only the outcomes consistent with the given information, then asks what fraction of that restricted set also satisfies the event in question.

The three setups below are the ones that appear most often, and each one has a specific trap built into its wording, which is exactly what makes them favourite opening questions in a first round interview.

At least one die shows a six. Eleven of the 36 outcomes have at least one six, six outcomes where the first die is six, six where the second die is six, minus the one outcome where both dice are six, counted twice otherwise. Only one of those eleven outcomes has both dice showing six, giving a conditional probability of one over eleven.

The sum equals eight. Five outcomes sum to eight, two and six, three and five, four and four, five and three, six and two. Two of those five outcomes contain a three, giving a conditional probability of two over five.