Difference between revisions of "2022 AMC 10B Problems/Problem 23"
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We denote by <math>\tau</math> the last step Amelia moves. Thus, <math>\tau \in \left\{ 2, 3 \right\}</math>. | We denote by <math>\tau</math> the last step Amelia moves. Thus, <math>\tau \in \left\{ 2, 3 \right\}</math>. | ||
We have | We have | ||
| + | |||
<math></math> | <math></math> | ||
\begin{align*} | \begin{align*} | ||
| Line 36: | Line 37: | ||
& = \left( 1 - \frac{1}{2} \right)\left( 1 - \frac{1}{2} \right) | & = \left( 1 - \frac{1}{2} \right)\left( 1 - \frac{1}{2} \right) | ||
+ \left( 1 - \frac{1}{6} \right) \frac{1}{2} \\ | + \left( 1 - \frac{1}{6} \right) \frac{1}{2} \\ | ||
| − | & = \boxed{\textbf{(C) <math>\frac{2}{3}</math>}} | + | & = \boxed{\textbf{(C) <math>\frac{2}{3}</math>}} , |
\end{align*} | \end{align*} | ||
<math></math> | <math></math> | ||
| + | |||
where the second equation follows from the property that <math>\left\{ x_n \right\}</math> and <math>\left\{ t_n \right\}</math> are independent sequences, the third equality follows from the lemma above. | where the second equation follows from the property that <math>\left\{ x_n \right\}</math> and <math>\left\{ t_n \right\}</math> are independent sequences, the third equality follows from the lemma above. | ||
Revision as of 14:07, 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 
,
![\[ \Bbb P \left( y_1 + y_2 \leq 1 \right) = \frac{1}{2} . \]](http://latex.artofproblemsolving.com/9/c/a/9ca0d2902c5eb8db92cddab19e3ca4e92b982504.png)
For 
,
![\[ \Bbb P \left( y_1 + y_2 + y_3 \leq 1 \right) = \frac{1}{6} . \]](http://latex.artofproblemsolving.com/f/7/e/f7e0ecac9af7c72deadc3b60deae91f69748cb18.png)
Now, we solve this problem.
We denote by 
the last step Amelia moves. Thus, 
.
We have
$$ (Error compiling LaTeX. Unknown error_msg)
\begin{align*}
\Bbb P \left( \sum_{n=1}^\tau x_n > 1 \right)
& = \Bbb P \left( x_1 + x_2 > 1 \bigg| t_1 + t_2 > 1 \right)
\Bbb P \left( t_1 + t_2 > 1 \right) \\
& \hspace{1cm} + \Bbb P \left( x_1 + x_2 + x_3 > 1 \bigg| t_1 + t_2 \leq 1 \right)
\Bbb P \left( t_1 + t_2 \leq 1 \right) \\
& = \Bbb P \left( x_1 + x_2 > 1 \right)
\Bbb P \left( t_1 + t_2 > 1 \right)
+ \Bbb P \left( x_1 + x_2 + x_3 > 1 \right)
\Bbb P \left( t_1 + t_2 \leq 1 \right) \\
& = \left( 1 - \frac{1}{2} \right)\left( 1 - \frac{1}{2} \right)
+ \left( 1 - \frac{1}{6} \right) \frac{1}{2} \\
& = \boxed{\textbf{(C) 
}} ,
\end{align*}
$$ (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)
