Difference between revisions of "2022 AMC 10B Problems/Problem 23"
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<math>x</math> must be a number between <math>0</math> and <math>1</math> (non-inclusive), and it is clearly greater than <math>\frac{1}{2}</math>, because the probability of getting more than <math>\frac{3}{2}</math> in <math>3</math> turns is <math>\frac{1}{2}</math>. Thus, the answer must be between <math>\frac{1}{2}</math> and <math>\frac{3}{4}</math>, non-inclusive, so the answer is <math>\fbox{C}</math>. | <math>x</math> must be a number between <math>0</math> and <math>1</math> (non-inclusive), and it is clearly greater than <math>\frac{1}{2}</math>, because the probability of getting more than <math>\frac{3}{2}</math> in <math>3</math> turns is <math>\frac{1}{2}</math>. Thus, the answer must be between <math>\frac{1}{2}</math> and <math>\frac{3}{4}</math>, non-inclusive, so the answer is <math>\fbox{C}</math>. | ||
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+ | ~mathboy100 | ||
==Video Solution== | ==Video Solution== |
Revision as of 16:26, 17 November 2022
Solution
We use the following lemma to solve this problem.
Let be independent random variables that are uniformly distributed on
. Then for
,
For ,
Now, we solve this problem.
We denote by the last step Amelia moves. Thus,
.
We have
where the second equation follows from the property that and
are independent sequences, the third equality follows from the lemma above.
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 2 (Elimination)
There is a probability that Amelia is past
after
turn, so Amelia can only pass
after
turns or
turns. The probability of finishing in
turns is
(due to the fact that the probability of getting
is the same as the probability of getting
), and thus the probability of finishing in
turns is also
.
It is also clear that the probability of Amelia being past in
turns is equal to
.
Therefore, if is the probability that Amelia finishes if she takes three turns, our final probability is
.
must be a number between
and
(non-inclusive), and it is clearly greater than
, because the probability of getting more than
in
turns is
. Thus, the answer must be between
and
, non-inclusive, so the answer is
.
~mathboy100
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)