Matching faces

Topic: Random experiments and probability · more questions on this topic
‹ Fifty-four distinct birthdays Three cards in order ›

Two fair dice are thrown, and the ordered pair is recorded.

  1. Find the probability that the two faces agree.

  2. Find the same probability given that the sum of the faces is at most 4.

  1. The first face may be anything. The second matches it with probability 1/6. There are 6 matching pairs out of 36.

  2. Let A be the event that the faces agree, and let B be the event that the sum is at most 4. Then

B=\{(1,1),(1,2),(1,3),(2,1),(2,2),(3,1)\}, \qquad A\cap B=\{(1,1),(2,2)\}.

Every ordered pair is equally likely, so \mathbb{P}(A\mid B)=2/6=1/3.