exercise:A068776306: 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> (Banach's Matchbox<ref group="Notes" >W. Feller, '' Introduction to Probability Theory,'' vol. 1, p. 166.</ref>) A man carries in each of his two front pockets a box of matches originally containing <math>N</math> matches. Whenever he needs a mat...") |
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(Banach's Matchbox<ref group="Notes" >W. Feller, '' Introduction to Probability Theory,'' vol. 1, p. 166.</ref>) A man carries in each of his two front | |||
Introduction to Probability Theory,'' vol. 1, p. 166.</ref>) A man carries in each of his two front | |||
pockets a box of matches originally containing <math>N</math> matches. Whenever he needs a match, he chooses | pockets a box of matches originally containing <math>N</math> matches. Whenever he needs a match, he chooses | ||
a pocket at random and removes one from that box. One day he reaches into a pocket | a pocket at random and removes one from that box. One day he reaches into a pocket | ||
and finds the box empty. | and finds the box empty. | ||
<ul><li> Let <math>p_r</math> denote the probability that the other pocket contains <math>r</math> matches. | <ul style="list-style-type:lower-alpha"><li> Let <math>p_r</math> denote the probability that the other pocket contains <math>r</math> matches. | ||
Define a sequence of ''counter'' random variables as follows: Let | Define a sequence of ''counter'' random variables as follows: Let | ||
<math>X_i = 1</math> if the <math>i</math>th draw is from the left pocket, and 0 if it is from the right | <math>X_i = 1</math> if the <math>i</math>th draw is from the left pocket, and 0 if it is from the right | ||
Latest revision as of 18:18, 14 June 2024
(Banach's Matchbox[Notes 1]) A man carries in each of his two front pockets a box of matches originally containing [math]N[/math] matches. Whenever he needs a match, he chooses a pocket at random and removes one from that box. One day he reaches into a pocket and finds the box empty.
- Let [math]p_r[/math] denote the probability that the other pocket contains [math]r[/math] matches. Define a sequence of counter random variables as follows: Let [math]X_i = 1[/math] if the [math]i[/math]th draw is from the left pocket, and 0 if it is from the right pocket. Interpret [math]p_r[/math] in terms of [math]S_n = X_1 + X_2 +\cdots+ X_n[/math]. Find a binomial expression for [math]p_r[/math].
- Write a computer program to compute the [math]p_r[/math], as well as the probability that the other pocket contains at least [math]r[/math] matches, for [math]N = 100[/math] and [math]r[/math] from 0 to 50.
- Show that [math](N - r)p_r = (1/2)(2N + 1)p_{r + 1} - (1/2)(r + 1)p_{r + 1}[/math]\ .
- Evaluate [math]\sum_r p_r[/math].
- Use (c) and (d) to determine the expectation [math]E[/math] of the distribution [math]\{p_r\}[/math].
- Use Stirling's formula to obtain an approximation for [math]E[/math]. How many matches must each box contain to ensure a value of about 13 for the expectation [math]E[/math]? (Take [math]\pi = 22/7[/math].)
Notes