MathDB
Coin Flips to Choose Real Numbers

Source: 2019 AMC 10A #22 / 2019 AMC 12A #20

February 8, 2019
AMCAMC 10AMC 12probability2019 AMC 12A2019 AMC 10A2019 AMC

Problem Statement

Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval [0,1][0,1]. Two random numbers xx and yy are chosen independently in this manner. What is the probability that ∣x−y∣>12|x-y| > \tfrac{1}{2}?
<spanclass=′latex−bold′>(A)</span>13<spanclass=′latex−bold′>(B)</span>716<spanclass=′latex−bold′>(C)</span>12<spanclass=′latex−bold′>(D)</span>916<spanclass=′latex−bold′>(E)</span>23<span class='latex-bold'>(A)</span> \frac{1}{3} \qquad <span class='latex-bold'>(B)</span> \frac{7}{16} \qquad <span class='latex-bold'>(C)</span> \frac{1}{2} \qquad <span class='latex-bold'>(D)</span> \frac{9}{16} \qquad <span class='latex-bold'>(E)</span> \frac{2}{3}
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