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Consider two independent random variables X and Y with identical distributions. The variables X and Y take values 0, 1 and 2 with probabilities 1/2, 1/4 and 1/4 respectively. What is the conditional probability P(X + Y = 2 | X − Y = 0)?

1. 0
2. 1/16
3. 1/6
4. 1

Correct answer: 1/6

Solution

X-Y=0 means X=Y, with P=1/4+1/16+1/16=3/8. Within that, X+Y=2 requires X=Y=1 with P=1/16. The conditional probability is (1/16)/(3/8)=1/6.

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