2019 AMC 12A Problem 20

Problem 20 of 25HarderCounting & Probability

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].

Two random numbers

x

and

y

are chosen independently in this manner. What is the probability that

|x - y| \gt \dfrac{1}{2}?

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Solution

Each variable equals

0

with probability

\tfrac14,

equals

1

with probability

\tfrac14,

and is uniform on

[0, 1]

with probability

\tfrac12.

Considering the nine combinations of types: the pairs

(0, 1)

and

(1, 0)

each contribute

\tfrac{1}{16}.

Each of the four point-versus-uniform cases contributes

\tfrac{1}{16}.

The uniform-versus-uniform case contributes

\tfrac14 \cdot \tfrac14 = \tfrac{1}{16}.

The total is

\dfrac{2 + 4 + 1}{16} = \dfrac{7}{16}.

Thus, the correct answer is B.

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