Difference between revisions of "2022 AMC 10B Problems/Problem 12"
MRENTHUSIASM (talk | contribs) |
MRENTHUSIASM (talk | contribs) |
||
Line 5: | Line 5: | ||
==Solution== | ==Solution== | ||
− | 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 | + | 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> |
+ | |||
+ | Therefore, we have <math>1-\left(\frac56\right)^n>\frac12,</math> or <cmath>\left(\frac56\right)^n<\frac12.</cmath> Since, <math>\left(\frac56\right)^3>\frac12>\left(\frac56\right)^4,</math> the least integer <math>n</math> satisfying the inequality is <math>\boxed{\textbf{(C) } 4}.</math> | ||
~MRENTHUSIASM | ~MRENTHUSIASM |
Revision as of 18:14, 17 November 2022
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
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
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.