Difference between revisions of "2020 AMC 10A Problems/Problem 25"

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{{duplicate|[[2020 AMC 12A Problems|2020 AMC 12A #23]] and [[2020 AMC 10A Problems|2020 AMC 10A #25]]}}
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==Problem==
 
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly <math>7.</math> Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
 
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly <math>7.</math> Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
  
==Solution==
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<math>\textbf{(A) } \frac{7}{36} \qquad\textbf{(B) } \frac{5}{24} \qquad\textbf{(C) } \frac{2}{9} \qquad\textbf{(D) } \frac{17}{72} \qquad\textbf{(E) } \frac{1}{4}</math>
Consider the probability that rolling two dice gives a sum of <math>s</math>. There are <math>s - 1</math> pairs that satisfy this, namely <math>(1, s - 1), (2, s - 2), ..., (s - 1, 1)</math>, out of <math>6^2 = 36</math> possible pairs. Thus, the probability is <math>\frac{s - 1}{36}</math>.
 
  
Therefore, if one dice has a value of <math>a</math> and Jason rerolls the other two dice, then the probability of winning is <math>\frac{7 - a - 1}{36} = \frac{6 - a}{36}</math>.
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==Solution 1==
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Consider the probability that rolling two dice gives a sum of <math>s</math>, where <math>s \leq 7</math>. There are <math>s - 1</math> pairs that satisfy this, namely <math>(1, s - 1), (2, s - 2), ..., (s - 1, 1)</math>, out of <math>6^2 = 36</math> possible pairs. The probability is <math>\frac{s - 1}{36}</math>.
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Therefore, if one die has a value of <math>a</math> and Jason rerolls the other two dice, then the probability of winning is <math>\frac{7 - a - 1}{36} = \frac{6 - a}{36}</math>.
  
 
In order to maximize the probability of winning, <math>a</math> must be minimized. This means that if Jason rerolls two dice, he must choose the two dice with the maximum values.
 
In order to maximize the probability of winning, <math>a</math> must be minimized. This means that if Jason rerolls two dice, he must choose the two dice with the maximum values.
  
Thus, we can let <math>a \leq b \leq c</math> be the values of the three dice <math>A</math>, <math>B</math>, and <math>C</math>. Consider the case when <math>a + b < 7</math>. If <math>a + b + c = 7</math>, then we do not need to reroll any dice. Otherwise,
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Thus, we can let <math>a \leq b \leq c</math> be the values of the three dice, which we will call <math>A</math>, <math>B</math>, and <math>C</math> respectively. Consider the case when <math>a + b < 7</math>. If <math>a + b + c = 7</math>, then we do not need to reroll any dice. Otherwise,
If we reroll one dice, we can roll dice <math>C</math> in the hope that we get the value that makes the sum of the three dice 7. This happens with probability <math>\frac16</math>.
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if we reroll one die, we can roll dice <math>C</math> in the hope that we get the value that makes the sum of the three dice <math>7</math>. This happens with probability <math>\frac16</math>. If we reroll two dice, we will roll <math>B</math> and <math>C</math>, and the probability of winning is <math>\frac{6 - a}{36}</math>, as stated above.
If we reroll two dice, we will roll both <math>B</math> and <math>C</math> and the probability of winning is <math>\frac{6 - a}{36}</math>, as stated above.
 
  
However, <math>\frac16 > \frac{6 - a}{36}</math>, so rolling one dice is always better than rolling two dice if <math>a + b < 7</math>.
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However, <math>\frac16 > \frac{6 - a}{36}</math>, so rolling one die is always better than rolling two dice if <math>a + b < 7</math>.
  
Now consider the case where <math>a + b \geq 7</math>. Rerolling one dice will not help us win since the sum of the three dice will always be greater than 7. If we reroll two dice, the probability of winning is, once again, <math>\frac{6 - a}{36}</math>. To find the probability of winning if we reroll all three dice, we can let each dice have 1 dot and find the number of ways to distribute the remaining 4 dots. By Stars and Bars, there are <math>{6\choose2} = 15</math> ways to do this, making the probability of winning <math>\frac{15}{6^3} = \frac5{72}</math>.
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Now consider the case where <math>a + b \geq 7</math>. Rerolling one die will not help us win since the sum of the three dice will always be greater than <math>7</math>. If we reroll two dice, the probability of winning is, once again, <math>\frac{6 - a}{36}</math>. To find the probability of winning if we reroll all three dice, we can let each dice have <math>1</math> dot and find the number of ways to distribute the remaining <math>4</math> dots. By stars and bars, there are <math>{6\choose2} = 15</math> ways to do this, making the probability of winning <math>\frac{15}{6^3} = \frac5{72}</math>.
  
 
In order for rolling two dice to be more favorable than rolling three dice, <math>\frac{6 - a}{36} > \frac5{72} \rightarrow a \leq 3</math>.
 
In order for rolling two dice to be more favorable than rolling three dice, <math>\frac{6 - a}{36} > \frac5{72} \rightarrow a \leq 3</math>.
  
The possible triplets <math>(a, b, c)</math> that satisfy <math>a \leq 3</math> and <math>a + b \geq 7</math>, and the number of ways each one can be permuted, are
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Thus, rerolling two dice is optimal if and only if <math>a \leq 3</math> and <math>a + b \geq 7</math>. The possible triplets <math>(a, b, c)</math> that satisfy these conditions, and the number of ways they can be permuted, are
 
<math>(3, 4, 4) \rightarrow 3</math> ways.
 
<math>(3, 4, 4) \rightarrow 3</math> ways.
 
<math>(3, 4, 5) \rightarrow 6</math> ways.
 
<math>(3, 4, 5) \rightarrow 6</math> ways.
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<math>(1, 6, 6) \rightarrow 3</math> ways.
 
<math>(1, 6, 6) \rightarrow 3</math> ways.
  
There are <math>3 + 6 + 6 + 3 + 6 + 3 + 3 + 6 + 3 + 3 = 42</math> favorable permuations out of <math>6^3 = 216</math> possibilities, making the overall probability <math>\frac{42}{216} = \boxed{\textbf{(A) }\frac7{36}}</math>
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There are <math>3 + 6 + 6 + 3 + 6 + 3 + 3 + 6 + 3 + 3 = 42</math> ways in which rerolling two dice is optimal, out of <math>6^3 = 216</math> possibilities, Therefore, the probability that Jason will reroll two dice is <math>\frac{42}{216} = \boxed{\textbf{(A) }\frac7{36}}</math>
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==Solution 2==
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We count the numerator.
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Jason will pick up no dice if he already has a 7 as a sum. We need to assume he does not have a 7 to begin with.
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If Jason decides to pick up all the dice to re-roll, by the stars and bars rule{ ways to distribute, <math>{n+k-1 \choose k-1}</math>, there will be 2 bars and 4 stars(3 of them need to be guaranteed because a roll is at least 1) for a probability of <math>\frac{15}{216}=\frac{2.5}{36}</math>.
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If Jason picks up 2 dice and leaves a die showing <math>k</math>, he will need the other two to sum to <math>7-k</math>. This happens with probability <cmath>\frac{6-k}{36}</cmath> for integers <math>1 \leq k \leq 6</math>.
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If the roll is not 7, Jason will pick up exactly one die to re-roll if there can remain two other dice with sum less than 7, since this will give him a <math>\frac{1}{6}</math> chance which is a larger probability than all the cases unless he has a 7 to begin with.
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We have <cmath>\frac{1}{6} > \underline{\frac{5,4,3}{36}} > \frac{2.5}{36} > \frac{2,1,0}{36}.</cmath>
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We count the underlined part's frequency for the numerator without upsetting the probability greater than it.
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Let <math>a</math> be the roll we keep. We know <math>a</math> is at most 3 since 4 would cause Jason to pick up all the dice.
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When <math>a=1</math>, there are 3 choices for whether it is rolled 1st, 2nd, or 3rd, and in this case the other two rolls have to be at least 6(or he would have only picked up 1). This give <math>3 \cdot 1^{2} =3</math> ways.
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Similarly, <math>a=2</math> gives <math>3 \cdot 2^{2} =12</math> because the 2 can be rolled in 3 places and the other two rolls are at least 5.
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<math>a=3</math> gives <math>3 \cdot 3^{2} =27</math>.
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Summing together gives the numerator of 42.
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The denominator is <math>6^3=216</math>, so we have <math>\frac{42}{216}=\boxed{(A) \frac{7}{36}}</math>
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== Video Solution ==
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https://youtu.be/3W4jOpCiBx8
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2020|ab=A|num-b=24|after=Last Problem}}
 
{{AMC10 box|year=2020|ab=A|num-b=24|after=Last Problem}}
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{{AMC12 box|year=2020|ab=A|num-b=22|num-a=24}}
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[[Category: Introductory Combinatorics Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 15:51, 9 May 2020

The following problem is from both the 2020 AMC 12A #23 and 2020 AMC 10A #25, so both problems redirect to this page.

Problem

Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7.$ Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?

$\textbf{(A) } \frac{7}{36} \qquad\textbf{(B) } \frac{5}{24} \qquad\textbf{(C) } \frac{2}{9} \qquad\textbf{(D) } \frac{17}{72} \qquad\textbf{(E) } \frac{1}{4}$

Solution 1

Consider the probability that rolling two dice gives a sum of $s$, where $s \leq 7$. There are $s - 1$ pairs that satisfy this, namely $(1, s - 1), (2, s - 2), ..., (s - 1, 1)$, out of $6^2 = 36$ possible pairs. The probability is $\frac{s - 1}{36}$.

Therefore, if one die has a value of $a$ and Jason rerolls the other two dice, then the probability of winning is $\frac{7 - a - 1}{36} = \frac{6 - a}{36}$.

In order to maximize the probability of winning, $a$ must be minimized. This means that if Jason rerolls two dice, he must choose the two dice with the maximum values.

Thus, we can let $a \leq b \leq c$ be the values of the three dice, which we will call $A$, $B$, and $C$ respectively. Consider the case when $a + b < 7$. If $a + b + c = 7$, then we do not need to reroll any dice. Otherwise, if we reroll one die, we can roll dice $C$ in the hope that we get the value that makes the sum of the three dice $7$. This happens with probability $\frac16$. If we reroll two dice, we will roll $B$ and $C$, and the probability of winning is $\frac{6 - a}{36}$, as stated above.

However, $\frac16 > \frac{6 - a}{36}$, so rolling one die is always better than rolling two dice if $a + b < 7$.

Now consider the case where $a + b \geq 7$. Rerolling one die will not help us win since the sum of the three dice will always be greater than $7$. If we reroll two dice, the probability of winning is, once again, $\frac{6 - a}{36}$. To find the probability of winning if we reroll all three dice, we can let each dice have $1$ dot and find the number of ways to distribute the remaining $4$ dots. By stars and bars, there are ${6\choose2} = 15$ ways to do this, making the probability of winning $\frac{15}{6^3} = \frac5{72}$.

In order for rolling two dice to be more favorable than rolling three dice, $\frac{6 - a}{36} > \frac5{72} \rightarrow a \leq 3$.

Thus, rerolling two dice is optimal if and only if $a \leq 3$ and $a + b \geq 7$. The possible triplets $(a, b, c)$ that satisfy these conditions, and the number of ways they can be permuted, are $(3, 4, 4) \rightarrow 3$ ways. $(3, 4, 5) \rightarrow 6$ ways. $(3, 4, 6) \rightarrow 6$ ways. $(3, 5, 5) \rightarrow 3$ ways. $(3, 5, 6) \rightarrow 6$ ways. $(3, 6, 6) \rightarrow 3$ ways. $(2, 5, 5) \rightarrow 3$ ways. $(2, 5, 6) \rightarrow 6$ ways. $(2, 6, 6) \rightarrow 3$ ways. $(1, 6, 6) \rightarrow 3$ ways.

There are $3 + 6 + 6 + 3 + 6 + 3 + 3 + 6 + 3 + 3 = 42$ ways in which rerolling two dice is optimal, out of $6^3 = 216$ possibilities, Therefore, the probability that Jason will reroll two dice is $\frac{42}{216} = \boxed{\textbf{(A) }\frac7{36}}$

Solution 2

We count the numerator. Jason will pick up no dice if he already has a 7 as a sum. We need to assume he does not have a 7 to begin with. If Jason decides to pick up all the dice to re-roll, by the stars and bars rule{ ways to distribute, ${n+k-1 \choose k-1}$, there will be 2 bars and 4 stars(3 of them need to be guaranteed because a roll is at least 1) for a probability of $\frac{15}{216}=\frac{2.5}{36}$. If Jason picks up 2 dice and leaves a die showing $k$, he will need the other two to sum to $7-k$. This happens with probability \[\frac{6-k}{36}\] for integers $1 \leq k \leq 6$. If the roll is not 7, Jason will pick up exactly one die to re-roll if there can remain two other dice with sum less than 7, since this will give him a $\frac{1}{6}$ chance which is a larger probability than all the cases unless he has a 7 to begin with. We have \[\frac{1}{6} > \underline{\frac{5,4,3}{36}} > \frac{2.5}{36} > \frac{2,1,0}{36}.\] We count the underlined part's frequency for the numerator without upsetting the probability greater than it. Let $a$ be the roll we keep. We know $a$ is at most 3 since 4 would cause Jason to pick up all the dice. When $a=1$, there are 3 choices for whether it is rolled 1st, 2nd, or 3rd, and in this case the other two rolls have to be at least 6(or he would have only picked up 1). This give $3 \cdot 1^{2} =3$ ways. Similarly, $a=2$ gives $3 \cdot 2^{2} =12$ because the 2 can be rolled in 3 places and the other two rolls are at least 5. $a=3$ gives $3 \cdot 3^{2} =27$. Summing together gives the numerator of 42. The denominator is $6^3=216$, so we have $\frac{42}{216}=\boxed{(A) \frac{7}{36}}$

Video Solution

https://youtu.be/3W4jOpCiBx8

See Also

2020 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 24
Followed by
Last Problem
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions
2020 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 22
Followed by
Problem 24
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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