Conditional Probability: Speed-Solving Techniques (4)

Question

A die is rolled twice. Given that the sum of the two rolls is 7, what is the probability that the first roll was 2?

Solution — Step by Step

Final answer: $P(A | B) = 1/6$.

Why This Works

Conditional probability narrows the sample space. Once we know the sum is 7, only six outcomes are possible, and they are equally likely. Of these, only one has first roll 2 — so $1/6$.

This is the “reduce sample space” speed trick. For any conditional probability with a uniform sample space, just count outcomes in the conditioned event and how many of them satisfy the second condition.

Alternative Method (Speed Solve)

List the 6 outcomes that sum to 7. Count how many have first roll = 2. Answer = (count)/(6). Done in 10 seconds.

Common Mistake

Confusing $P(A | B)$ with $P(B | A)$. Here $P(B | A)$ would be “given first roll is 2, probability sum is 7”, which is the probability second roll is 5, i.e., also $1/6$. They happen to be equal here by coincidence — usually they differ.