Difference between revisions of "2022 AMC 10B Problems/Problem 12"
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~MRENTHUSIASM | ~MRENTHUSIASM | ||
− | ==Solution 2 (99% | + | ==Solution 2 (99% Accurate Guesswork)== |
− | Let's try the answer choices. We can quickly find that when we roll <math>3</math> dice, either the first and second sum to <math>7</math>, the first and third sum to <math>7</math>, or the second and third sum to <math>7</math>. There are <math>6</math> ways for the first and second dice to sum to <math>7</math>, <math>6</math> ways for the first and third to sum to <math>7</math>, and <math>6</math> ways for the second and third dice to sum to <math>7</math>. However, we overcounted (but not by much) so we can assume that the answer is <math>\boxed {\textbf{(C) 4 | + | Let's try the answer choices. We can quickly find that when we roll <math>3</math> dice, either the first and second sum to <math>7</math>, the first and third sum to <math>7</math>, or the second and third sum to <math>7</math>. There are <math>6</math> ways for the first and second dice to sum to <math>7</math>, <math>6</math> ways for the first and third to sum to <math>7</math>, and <math>6</math> ways for the second and third dice to sum to <math>7</math>. However, we overcounted (but not by much) so we can assume that the answer is <math>\boxed {\textbf{(C) }4}</math> |
~Arcticturn | ~Arcticturn |
Revision as of 08:33, 19 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
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
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.