The die came up even, is it a four?

Question

You roll one fair six-sided die but do not see it. A friend glances and tells you the result is an even number. Given that, what is the probability the die shows a four?

Accepted Answer

Before the clue, the possible results are \{1,2,3,4,5,6\}. The clue "even" keeps only: \{2,\ 4,\ 6\}. These three are still equally likely, and exactly one of them is a four: P(	ext{four} \mid 	ext{even}) = \frac{1}{3}. Conditioning just throws away the outcomes that clash with the clue (the odd numbers) and re-scales the rest so they sum to one. Takeaway. A conditional probability is a fraction over the reduced sample space: count the outcomes that fit both the clue and the target, divided by the outcomes that fit the clue.

The die came up even, is it a four?

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