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==Solution 1== | ==Solution 1== | ||
− | 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), | + | 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), \ldots, (s - 1, 1)</math>, out of <math>6^2 = 36</math> possible pairs. The probability is <math>\frac{s - 1}{36}</math>. |
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>. | 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>. | ||
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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 | + | Thus, we can let <math>a \leq b \leq c</math> be the values of the three dice. 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 die, we can | + | if we reroll one die, we can reroll any one dice 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 the two maximum dice, 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 die is always better than rolling two dice if <math>a + b < 7</math>. |
− | |||
− | 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 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>. | 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>. | ||
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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 | 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. | ||
+ | |||
<math>(3, 4, 6) \rightarrow 6</math> ways. | <math>(3, 4, 6) \rightarrow 6</math> ways. | ||
+ | |||
<math>(3, 5, 5) \rightarrow 3</math> ways. | <math>(3, 5, 5) \rightarrow 3</math> ways. | ||
+ | |||
<math>(3, 5, 6) \rightarrow 6</math> ways. | <math>(3, 5, 6) \rightarrow 6</math> ways. | ||
+ | |||
<math>(3, 6, 6) \rightarrow 3</math> ways. | <math>(3, 6, 6) \rightarrow 3</math> ways. | ||
+ | |||
<math>(2, 5, 5) \rightarrow 3</math> ways. | <math>(2, 5, 5) \rightarrow 3</math> ways. | ||
+ | |||
<math>(2, 5, 6) \rightarrow 6</math> ways. | <math>(2, 5, 6) \rightarrow 6</math> ways. | ||
+ | |||
<math>(2, 6, 6) \rightarrow 3</math> ways. | <math>(2, 6, 6) \rightarrow 3</math> ways. | ||
+ | |||
<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> 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) }\ | + | 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) } \frac{7}{36}}</math>. |
+ | |||
+ | ~edits by eagleye | ||
+ | |||
+ | ==Solution 2 (Illustration)== | ||
+ | We conclude all of the following after the initial roll: | ||
+ | <ol style="margin-left: 1.5em;"> | ||
+ | <li>Jason rerolls exactly zero dice if and only if the sum of the three dice is <math>7,</math> in which the probability of winning is always <math>1.</math></li><p> | ||
+ | <li>If Jason rerolls exactly one die, then the sum of the two other dice must be <math>2,3,4,5,</math> or <math>6.</math> The probability of winning is always <math>\frac16,</math> as exactly <math>1</math> of the <math>6</math> possible outcomes of the die rerolled results in a win. </li><p> | ||
+ | <li>If Jason rerolls exactly two dice, then the outcome of the remaining die must be <math>1,2,3,4,</math> or <math>5.</math> Applying casework to the remaining die produces the following table: | ||
+ | <cmath>\begin{array}{c||c|c} | ||
+ | & & \\ [-2.5ex] | ||
+ | \textbf{Remaining Die} & \textbf{Sum Needed for the Two Dice Rerolled} & \textbf{Probability of Winning} \\ [0.5ex] | ||
+ | \hline | ||
+ | & & \\ [-2ex] | ||
+ | 1 & 6 & 5/36 \\ | ||
+ | 2 & 5 & 4/36 \\ | ||
+ | 3 & 4 & 3/36 \\ | ||
+ | 4 & 3 & 2/36 \\ | ||
+ | 5 & 2 & 1/36 | ||
+ | \end{array}</cmath> | ||
+ | The probability of winning is at most <math>\frac{5}{36}.</math></li><p> | ||
+ | <li>If Jason (re)rolls all three dice, then the probability of winning is always <math>\frac{\binom62}{6^3}=\frac{15}{216}=\frac{5}{72}.</math></li><p> | ||
+ | For the denominator, rolling three dice gives a total of <math>6^3=216</math> possible outcomes. <p> | ||
+ | For the numerator, this is the same as counting the ordered triples of positive integers <math>(a,b,c)</math> for which <math>a+b+c=7.</math> Suppose that <math>7</math> balls are lined up in a row. There are <math>6</math> gaps between the balls, and placing dividers in <math>2</math> of the gaps separates the balls into <math>3</math> piles. From left to right, the numbers of balls in the piles correspond to <math>a,b,</math> and <math>c,</math> respectively. There are <math>\binom62=15</math> ways to place the dividers. Note that the dividers' positions and the ordered triples have one-to-one correspondence, and <math>1\leq a,b,c\leq6</math> holds for all such ordered triples. </li><p> | ||
+ | </ol> | ||
+ | The optimal strategy is that: | ||
+ | <ul style="list-style-type:square;"> | ||
+ | <li>If Jason needs to reroll at least zero dice to win, then he rerolls exactly zero dice.</li><p> | ||
+ | <li>If Jason needs to reroll at least one die to win, then he rerolls exactly one die.</li><p> | ||
+ | <li>If Jason needs to reroll at least two dice to win, then he rerolls exactly two dice if and only if the probability of winning is greater than <math>\frac{5}{72},</math> the probability of winning for rerolling all three dice. <p> The first three cases in the table above satisfy this requirement. We will analyze these cases by considering the initial outcomes of the two dice rerolled. Note that any of the three dice can be the remaining die, so we need a factor of <math>3</math> for all counts in the third column: | ||
+ | <cmath>\begin{array}{c||c|c} | ||
+ | & & \\ [-2.5ex] | ||
+ | \textbf{Remaining Die} & \textbf{Initial Outcomes of the Two Dice Rerolled} & \textbf{\# of Ways} \\ [0.5ex] | ||
+ | \hline | ||
+ | & & \\ [-2ex] | ||
+ | 1 & \text{Both are in }\{6\}\text{, not necessarily distinct.} & 3\cdot1^2=3 \\ | ||
+ | 2 & \text{Both are in }\{5,6\}\text{, not necessarily distinct.} & \hspace{2mm}3\cdot2^2=12 \\ | ||
+ | 3 & \text{Both are in }\{4,5,6\}\text{, not necessarily distinct.} & \hspace{2mm}3\cdot3^2=27 | ||
+ | \end{array}</cmath> | ||
+ | </li> | ||
+ | </ul> | ||
+ | Finally, the requested probability is <cmath>\frac{3+12+27}{6^3}=\frac{42}{216}=\boxed{\textbf{(A) } \frac{7}{36}}.</cmath> | ||
+ | |||
+ | ~MRENTHUSIASM | ||
− | ==Solution 2== | + | ==Solution 3 (Similar to Solution 2)== |
We count the numerator. | 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. | + | Jason will pick up no dice if he already has a <math>7</math> as a sum. We need to assume he does not have a <math>7</math> to begin with. |
− | If Jason decides to pick up all the dice to re-roll, by the stars and bars rule | + | 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 <math>2</math> bars and <math>4</math> stars (<math>3</math> of them need to be guaranteed because a roll is at least <math>1</math>) for a probability of <math>\frac{15}{216}=\frac{2.5}{36}</math>. |
− | 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>. | + | If Jason picks up <math>2</math> 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>. |
− | 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. | + | If the roll is not <math>7</math>, Jason will pick up exactly one die to re-roll if there can remain two other dice with sum less than <math>7</math>, 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 <math>7</math> to begin with. |
We have <cmath>\frac{1}{6} > \underline{\frac{5,4,3}{36}} > \frac{2.5}{36} > \frac{2,1,0}{36}.</cmath> | We have <cmath>\frac{1}{6} > \underline{\frac{5,4,3}{36}} > \frac{2.5}{36} > \frac{2,1,0}{36}.</cmath> | ||
We count the underlined part's frequency for the numerator without upsetting the probability greater than it. | We count the underlined part's frequency for the numerator without upsetting the probability greater than it. | ||
− | Let <math>a</math> be the roll we keep. We know <math>a</math> | + | Let <math>a</math> be the roll we keep. We know <math>a\leq3</math> since <math>a=4</math> would cause Jason to pick up all the dice. |
− | When <math>a=1</math>, there are 3 choices for whether it is rolled | + | When <math>a=1</math>, there are <math>3</math> choices for whether it is rolled <math>1</math>st, <math>2</math>nd, or <math>3</math>rd, and in this case the other two rolls have to be at least <math>6</math> (or he would have only picked up <math>1</math>). This give <math>3 \cdot 1^{2} =3</math> ways. |
− | 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. | + | Similarly, <math>a=2</math> gives <math>3 \cdot 2^{2} =12</math> because the <math>2</math> can be rolled in <math>3</math> places and the other two rolls are at least <math>5</math>. |
<math>a=3</math> gives <math>3 \cdot 3^{2} =27</math>. | <math>a=3</math> gives <math>3 \cdot 3^{2} =27</math>. | ||
− | Summing together gives the numerator of 42. | + | Summing together gives the numerator of <math>42</math>. |
− | The denominator is <math>6^3=216</math>, so we have <math>\frac{42}{216}=\boxed{(A) \frac{7}{36}}</math> | + | The denominator is <math>6^3=216</math>, so we have <math>\frac{42}{216}=\boxed{\textbf{(A) } \frac{7}{36}}</math>. |
+ | |||
+ | ==Solution 4== | ||
+ | Note that Jason will roll <math>2</math> dies if and only if the pairwise sum of the dies is greater than <math>7</math>. Suppose that Jason doesn't roll a <math>1</math>. This means that roll was <math>(1, 6, 6)</math> which can be arranged in <math>3</math> ways. Suppose that Jason doesn't roll a <math>2</math>. This means the roll was either <math>(2, 5/6, 5/6)</math> which can be arranged in <math>12</math> ways. Suppose that Jason doesn't roll a <math>3</math>. This means that the roll was either <math>(3, 4/5/6, 4/5/6)</math> which can be arranged in <math>27</math> ways. Jason can't roll a <math>4, 5, 6</math> because he can't get a pairwise sum of <math>7</math> from those numbers. Thus he rolls two dies exactly <math>27 + 12 + 3 = 42</math> out of the <math>6^3</math> possible rolls. So the answer is <math>\frac{42}{216} = \boxed{\textbf{(A) } \frac{7}{36}}</math>. | ||
+ | |||
+ | ~coolmath_2018 | ||
− | == Video Solution == | + | ==Video Solutions== |
+ | ===Video Solution 1=== | ||
+ | https://youtu.be/B8pt8jF04ZM - Happytwin | ||
− | https:// | + | ===Video Solution 2 (Richard Rusczyk)=== |
+ | https://artofproblemsolving.com/videos/amc/2020amc10a/516 | ||
==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}} | ||
{{AMC12 box|year=2020|ab=A|num-b=22|num-a=24}} | {{AMC12 box|year=2020|ab=A|num-b=22|num-a=24}} | ||
+ | |||
+ | [[Category: Introductory Combinatorics Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 06:10, 26 October 2021
- The following problem is from both the 2020 AMC 12A #23 and 2020 AMC 10A #25, so both problems redirect to this page.
Contents
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 Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
Solution 1
Consider the probability that rolling two dice gives a sum of , where . There are pairs that satisfy this, namely , out of possible pairs. The probability is .
Therefore, if one die has a value of and Jason rerolls the other two dice, then the probability of winning is .
In order to maximize the probability of winning, 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 be the values of the three dice. Consider the case when . If , then we do not need to reroll any dice. Otherwise, if we reroll one die, we can reroll any one dice in the hope that we get the value that makes the sum of the three dice . This happens with probability . If we reroll two dice, we will roll the two maximum dice, and the probability of winning is , as stated above. However, , so rolling one die is always better than rolling two dice if .
Now consider the case where . Rerolling one die will not help us win since the sum of the three dice will always be greater than . If we reroll two dice, the probability of winning is, once again, . To find the probability of winning if we reroll all three dice, we can let each dice have dot and find the number of ways to distribute the remaining dots. By stars and bars, there are ways to do this, making the probability of winning .
In order for rolling two dice to be more favorable than rolling three dice, .
Thus, rerolling two dice is optimal if and only if and . The possible triplets that satisfy these conditions, and the number of ways they can be permuted, are
ways.
ways.
ways.
ways.
ways.
ways.
ways.
ways.
ways.
ways.
There are ways in which rerolling two dice is optimal, out of possibilities, Therefore, the probability that Jason will reroll two dice is .
~edits by eagleye
Solution 2 (Illustration)
We conclude all of the following after the initial roll:
- Jason rerolls exactly zero dice if and only if the sum of the three dice is in which the probability of winning is always
- If Jason rerolls exactly one die, then the sum of the two other dice must be or The probability of winning is always as exactly of the possible outcomes of the die rerolled results in a win.
- If Jason rerolls exactly two dice, then the outcome of the remaining die must be or Applying casework to the remaining die produces the following table: The probability of winning is at most
- If Jason (re)rolls all three dice, then the probability of winning is always
For the denominator, rolling three dice gives a total of possible outcomes.
For the numerator, this is the same as counting the ordered triples of positive integers for which Suppose that balls are lined up in a row. There are gaps between the balls, and placing dividers in of the gaps separates the balls into piles. From left to right, the numbers of balls in the piles correspond to and respectively. There are ways to place the dividers. Note that the dividers' positions and the ordered triples have one-to-one correspondence, and holds for all such ordered triples.
The optimal strategy is that:
- If Jason needs to reroll at least zero dice to win, then he rerolls exactly zero dice.
- If Jason needs to reroll at least one die to win, then he rerolls exactly one die.
- If Jason needs to reroll at least two dice to win, then he rerolls exactly two dice if and only if the probability of winning is greater than the probability of winning for rerolling all three dice.
The first three cases in the table above satisfy this requirement. We will analyze these cases by considering the initial outcomes of the two dice rerolled. Note that any of the three dice can be the remaining die, so we need a factor of for all counts in the third column:
Finally, the requested probability is
~MRENTHUSIASM
Solution 3 (Similar to Solution 2)
We count the numerator. Jason will pick up no dice if he already has a as a sum. We need to assume he does not have a to begin with. If Jason decides to pick up all the dice to re-roll, by the stars and bars rule ways to distribute, , there will be bars and stars ( of them need to be guaranteed because a roll is at least ) for a probability of . If Jason picks up dice and leaves a die showing , he will need the other two to sum to . This happens with probability for integers . If the roll is not , Jason will pick up exactly one die to re-roll if there can remain two other dice with sum less than , since this will give him a chance which is a larger probability than all the cases unless he has a to begin with. We have We count the underlined part's frequency for the numerator without upsetting the probability greater than it. Let be the roll we keep. We know since would cause Jason to pick up all the dice. When , there are choices for whether it is rolled st, nd, or rd, and in this case the other two rolls have to be at least (or he would have only picked up ). This give ways. Similarly, gives because the can be rolled in places and the other two rolls are at least . gives . Summing together gives the numerator of . The denominator is , so we have .
Solution 4
Note that Jason will roll dies if and only if the pairwise sum of the dies is greater than . Suppose that Jason doesn't roll a . This means that roll was which can be arranged in ways. Suppose that Jason doesn't roll a . This means the roll was either which can be arranged in ways. Suppose that Jason doesn't roll a . This means that the roll was either which can be arranged in ways. Jason can't roll a because he can't get a pairwise sum of from those numbers. Thus he rolls two dies exactly out of the possible rolls. So the answer is .
~coolmath_2018
Video Solutions
Video Solution 1
https://youtu.be/B8pt8jF04ZM - Happytwin
Video Solution 2 (Richard Rusczyk)
https://artofproblemsolving.com/videos/amc/2020amc10a/516
See Also
2020 AMC 10A (Problems • Answer Key • Resources) | ||
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 (Problems • Answer Key • Resources) | |
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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