exercise:810f86be70: Difference between revisions
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(Created page with "<div class="d-none"><math> \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}</math></div> A coin is in one of <math>n</math> boxes. The probability that it is in the <math>i</math>th box is <math>p_i</math>. If you search in the <math>i</math>th box and it is there, you find it with probability <math>a_i</math>. Show that the probab...") |
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The probability that it is in the <math>i</math>th box is <math>p_i</math>. If you search in the <math>i</math>th box and it is there, you find it with | |||
box is <math>p_i</math>. If you search in the <math>i</math>th box and it is there, you find it with | |||
probability <math>a_i</math>. Show that the probability <math>p</math> that the coin is in the <math>j</math>th | probability <math>a_i</math>. Show that the probability <math>p</math> that the coin is in the <math>j</math>th | ||
box, given that you have looked in the <math>i</math>th box and not found it, is | box, given that you have looked in the <math>i</math>th box and not found it, is | ||
Latest revision as of 00:18, 13 June 2024
The probability that it is in the [math]i[/math]th box is [math]p_i[/math]. If you search in the [math]i[/math]th box and it is there, you find it with probability [math]a_i[/math]. Show that the probability [math]p[/math] that the coin is in the [math]j[/math]th box, given that you have looked in the [math]i[/math]th box and not found it, is
[[math]]
p = \left \{ \matrix{
p_j/(1-a_ip_i),&\,\,\, \mbox{if} \,\,\, j \ne i,\cr
(1 - a_i)p_i/(1 - a_ip_i),&\,\,\,\mbox{if} \,\, j = i.\cr}\right.
[[/math]]