Difference between revisions of "2022 AMC 10B Problems/Problem 12"
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<math>\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 5 \qquad \textbf{(E) } 6</math> | <math>\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 5 \qquad \textbf{(E) } 6</math> | ||
− | ==Solution== | + | ==Solution 1== |
Rolling a pair of fair <math>6</math>-sided dice, the probability of getting a sum of <math>7</math> is <math>\frac16:</math> Regardless what the first die shows, the second die has exactly one outcome to make the sum <math>7.</math> We consider the complement: The probability of not getting a sum of <math>7</math> is <math>1-\frac16=\frac56.</math> Rolling the pair of dice <math>n</math> times, the probability of getting a sum of <math>7</math> at least once is <math>1-\left(\frac56\right)^n.</math> | Rolling a pair of fair <math>6</math>-sided dice, the probability of getting a sum of <math>7</math> is <math>\frac16:</math> Regardless what the first die shows, the second die has exactly one outcome to make the sum <math>7.</math> We consider the complement: The probability of not getting a sum of <math>7</math> is <math>1-\frac16=\frac56.</math> Rolling the pair of dice <math>n</math> times, the probability of getting a sum of <math>7</math> at least once is <math>1-\left(\frac56\right)^n.</math> | ||
Revision as of 19:18, 11 January 2023
Problem
A pair of fair -sided dice is rolled times. What is the least value of such that the probability that the sum of the numbers face up on a roll equals at least once is greater than ?
Solution 1
Rolling a pair of fair -sided dice, the probability of getting a sum of is Regardless what the first die shows, the second die has exactly one outcome to make the sum We consider the complement: The probability of not getting a sum of is Rolling the pair of dice times, the probability of getting a sum of at least once is
Therefore, we have or Since the least integer satisfying the inequality is
~MRENTHUSIASM
Solution 2 (99% Accurate Guesswork)
Let's try the answer choices. We can quickly find that when we roll dice, either the first and second sum to , the first and third sum to , or the second and third sum to . There are ways for the first and second dice to sum to , ways for the first and third to sum to , and ways for the second and third dice to sum to . However, we overcounted (but not by much) so we can assume that the answer is
~Arcticturn
Video Solution 1
~Education, the Study of Everything
See Also
2022 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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 |
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