2022 AMC 10B Problems/Problem 23

Solution

We use the following lemma to solve this problem.

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Let be independent random variables that are uniformly distributed on . Then for ,

For ,

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Now, we solve this problem.

We denote by the last step Amelia moves. Thus, . We have

\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) \frac{2}{3}}} ,
\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

https://youtu.be/qOxnx_c9kVo

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)