Difference between revisions of "2021 AMC 10B Problems/Problem 18"
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| − | + | ==Problem== | |
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| + | A fair <math>6</math>-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number? | ||
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| + | <math>\textbf{(A)} ~\frac{1}{120} \qquad\textbf{(B)} ~\frac{1}{32} \qquad\textbf{(C)} ~\frac{1}{20} \qquad\textbf{(D)} ~\frac{3}{20} \qquad\textbf{(E)} ~\frac{1}{6}</math> | ||
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| + | ==Solution== | ||
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| + | There is a <math>\frac{3}6</math> chance that the first number we choose is even. | ||
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| + | There is a <math>\frac{2}5</math> chance that the next number that is distinct from the first is even. | ||
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| + | There is a <math>\frac{1}4</math> chance that the next number distinct from the first two is even. | ||
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| + | <math>\frac{3}6 * \frac{2}5 * \frac{1}4 = \frac{1}{20}</math>, so the answer is <math> \boxed{ C) \frac{1}{20} }</math> | ||
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| + | ~Tucker | ||
Revision as of 18:24, 11 February 2021
Problem
A fair 
-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?
Solution
There is a 
chance that the first number we choose is even.
There is a 
chance that the next number that is distinct from the first is even.
There is a 
chance that the next number distinct from the first two is even.
, so the answer is 
~Tucker
