2018 AMC 12A Problems/Problem 24

Problem

Alice, Bob, and Carol play a game in which each of them chooses a real number between $0$ and $1.$ The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between $0$ and $1,$ and Bob announces that he will choose his number uniformly at random from all the numbers between $\tfrac{1}{2}$ and $\tfrac{2}{3}.$ Armed with this information, what number should Carol choose to maximize her chance of winning?

$\textbf{(A) }\frac{1}{2}\qquad \textbf{(B) }\frac{13}{24} \qquad \textbf{(C) }\frac{7}{12} \qquad \textbf{(D) }\frac{5}{8} \qquad \textbf{(E) }\frac{2}{3}\qquad$

Solution 1 (Answer Choices)

Let $a,b,$ and $c$ be the numbers that Alice, Bob, and Carol choose, respectively.

From the answer choices, we construct the following table: $\begin{array}{c|c|c} & & \\ [-2ex] \boldsymbol{c} & \textbf{Conditions for }\boldsymbol{a}\textbf{ and }\boldsymbol{b} & \textbf{Carol's Probability of Winning} \\ [0.5ex] \hline & & \\ [-1.5ex] \frac12 & 0<a<c \text{ and } c<b<\frac23 & \hspace{21.625mm}\frac{c}{1}\cdot\frac{2/3-c}{1/6}=\frac12 \\ [1.5ex] \frac{13}{24} & \left(0<a<c \text{ and } c<b<\frac23\right) \text{ or } \left(c<a<1 \text{ and } \frac12<b<c\right) & \hspace{1.25mm}\frac{c}{1}\cdot\frac{2/3-c}{1/6}+\frac{1-c}{1}\cdot\frac{c-1/2}{1/6}=\frac{25}{48} \\ [1.5ex] \frac{7}{12} & \left(0<a<c \text{ and } c<b<\frac23\right) \text{ or } \left(c<a<1 \text{ and } \frac12<b<c\right) & \frac{c}{1}\cdot\frac{2/3-c}{1/6}+\frac{1-c}{1}\cdot\frac{c-1/2}{1/6}=\frac12 \\ [1.5ex] \frac58 & \left(0<a<c \text{ and } c<b<\frac23\right) \text{ or } \left(c<a<1 \text{ and } \frac12<b<c\right) & \hspace{1.25mm}\frac{c}{1}\cdot\frac{2/3-c}{1/6}+\frac{1-c}{1}\cdot\frac{c-1/2}{1/6}=\frac{7}{16} \\ [1.5ex] \frac23 & c<a<1 \text{ and } \frac12<b<c & \hspace{18.25mm}\frac{1-c}{1}\cdot\frac{c-1/2}{1/6}=\frac13 \\ [1.5ex] \end{array}$ Therefore, Carol should choose $\boxed{\textbf{(B) }\frac{13}{24}}$ to maximize her chance of winning.

~MRENTHUSIASM

Solution 2 (Piecewise Function)

Let $a,b,$ and $c$ be the numbers that Alice, Bob, and Carol choose, respectively.

Based on the value of $c,$ we construct the following table: $\begin{array}{c|c|c} & & \\ [-2ex] \textbf{Case} & \textbf{Conditions for }\boldsymbol{a}\textbf{ and }\boldsymbol{b} & \textbf{Carol's Probability of Winning} \\ [0.5ex] \hline & & \\ [-1.5ex] 0<c<\frac12 & 0<a<c \text{ and } c<b<\frac23 & \hspace{12.5mm}\frac{c}{1}\cdot\frac{2/3-c}{1/6}=-6c^2+4c \\ [1.5ex] \frac12\leq c<\frac23 & \left(0<a<c \text{ and } c<b<\frac23\right) \text{ or } \left(c<a<1 \text{ and } \frac12<b<c\right) & \hspace{1.25mm}\frac{c}{1}\cdot\frac{2/3-c}{1/6}+\frac{1-c}{1}\cdot\frac{c-1/2}{1/6}=-12c^2+13c-3 \\ [1.5ex] \frac23\leq c<1 & c<a<1 \text{ and } \frac12<b<c & \hspace{15.75mm}\frac{1-c}{1}\cdot\frac{c-1/2}{1/6}=-6c^2+9c-3 \\ [1.5ex] \end{array}$ Let $P(c)$ be Carol's probability of winning when she chooses $c.$ We write $P(c)$ as a piecewise function: $P(c) = \begin{cases} -6c^2+4c & \mathrm{if} \ x \le 0 \\ -12c^2+13c-3 & \mathrm{if} \ x > 0 \end{cases}.$

Solution 3 (5-second way)

The expected value of Alice's number is $\frac{1}{2}$ and the expected value of Bob's number is $\frac{7}{12}$ . To maximize her chance of winning, Carol would choose number exactly in between the two expected values, giving: $\frac{6+7}{12*2}=\frac{13}{24}$ . This is $\boxed{\textbf{(B)}}$ . (Random_Guy)

Even faster, once you know it's between $\frac{1}{2}$ and $\frac{7}{12},$ the answer is $B$ because no other option is in this interval.

Solution 4

Let’s call Alice’s number a, Bob’s number b, and Carol’s number c. Then, in order to maximize her chance of choosing a number that is in between a and b, she should choose c = (a+b)/2.

We need to find the average value of (a+b)/2 over the region [0, 1] x[1/2, 2/3] in the a-b plane.

We can set up a double integral with bounds 0 to 1 for the outer integral and 1/2 to 2/3 for the inner integral with an integrand of (a+b)/2. We need to divide our answer by 1/6, the area of the region of interest. This should yield 13/24, B.

Solution 5

Have Carol pick a point $x$ on the real line. The probability that Alice's number is below is thus $x$ , and the probability that Alice's number is above is $1 - x$ . Now, Carol must pick a point between $\frac{1}{2}$ and $\frac{2}{3}$ , exclusive. If she does not, then it is impossible for Carol's point to be between both Alice and Bob's point. Now what is the probability that Bob's point is below Carol's? To make it simple, scale Bob's number to the normal number line by subtracting $\frac{1}{2}$ and multiplying by $6$ . So thus the chance Bob's number is below is $6(x - \frac{1}{2})$ and above is $1 - 6(x-\frac{1}{2})$ .

Now we want to maximixe the probability of (Alice number below $x$ ) AND (Bob number above $x$ ) + (Alice number above $x$ ) AND (Bob number below $x$ ). This gives the formula $x(1 - 6(x-\frac{1}{2})) + (1 -x)6(x-\frac{1}{2}) = -3 + 13 x - 12 x^2$ . Then proceed using the process of completing the square or averaging the roots to get that the maximum of this is $\frac{13}{24}$ .

Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2018amc12a/474

~ dolphin7

Video Solution (Meta-Solving Technique)

https://youtu.be/GmUWIXXf_uk?t=926

~ pi_is_3.14

2018 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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