Difference between revisions of "2022 AMC 10B Problems/Problem 23"
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\begin{align*} | \begin{align*} | ||
P \left( \sum_{n=1}^\tau x_n > 1 \right) | P \left( \sum_{n=1}^\tau x_n > 1 \right) | ||
− | & = P \left( x_1 + x_2 > 1 | + | & = P \left( x_1 + x_2 > 1 | t_1 + t_2 > 1 \right) |
P \left( t_1 + t_2 > 1 \right) \ | P \left( t_1 + t_2 > 1 \right) \ | ||
− | & \hspace{1cm} + P \left( x_1 + x_2 + x_3 > 1 | + | & \hspace{1cm} + P \left( x_1 + x_2 + x_3 > 1 | t_1 + t_2 \leq 1 \right) |
P \left( t_1 + t_2 \leq 1 \right) \ | P \left( t_1 + t_2 \leq 1 \right) \ | ||
& = P \left( x_1 + x_2 > 1 \right) | & = P \left( x_1 + x_2 > 1 \right) |
Revision as of 14:08, 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
$$ (Error compiling LaTeX. Unknown error_msg) \begin{align*} P \left( \sum_{n=1}^\tau x_n > 1 \right) & = P \left( x_1 + x_2 > 1 | t_1 + t_2 > 1 \right)
P \left( t_1 + t_2 > 1 \right) \\
& \hspace{1cm} + P \left( x_1 + x_2 + x_3 > 1 | t_1 + t_2 \leq 1 \right)
P \left( t_1 + t_2 \leq 1 \right) \
& = P \left( x_1 + x_2 > 1 \right)
P \left( t_1 + t_2 > 1 \right)
+ P \left( x_1 + x_2 + x_3 > 1 \right)
P \left( t_1 + t_2 \leq 1 \right) \
& = \left( 1 - \frac{1}{2} \right)\left( 1 - \frac{1}{2} \right)
+ \left( 1 - \frac{1}{6} \right) \frac{1}{2} \
& = \boxed{\textbf{(C) }} ,
\end{align*}
$$ (Error compiling LaTeX. Unknown error_msg)
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)
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)