exercise:3603a3b6c0: 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> You are playing ''heads or tails'' with Prosser but you suspect that his coin is unfair. Von Neumann suggested that you proceed as follows: Toss Prosser's coin twice. If the outcome is HT call the result ''win.'' if it is TH call the result ''lo...") |
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You are playing ''heads or tails'' with Prosser but you suspect that his coin is unfair. Von Neumann suggested that you proceed as follows: | |||
suspect that his coin is unfair. Von Neumann suggested that you proceed as follows: | |||
Toss Prosser's coin twice. If the outcome is HT call the result ''win.'' if it is | Toss Prosser's coin twice. If the outcome is HT call the result ''win.'' if it is | ||
TH call the result ''lose.'' If it is TT or HH ignore the outcome and toss | TH call the result ''lose.'' If it is TT or HH ignore the outcome and toss | ||
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the result win or lose in a single play. Repeat this procedure for each play. Assume | the result win or lose in a single play. Repeat this procedure for each play. Assume | ||
that Prosser's coin turns up heads with probability <math>p</math>. | that Prosser's coin turns up heads with probability <math>p</math>. | ||
<ul><li> Find the probability of HT, TH, HH, TT with two tosses of Prosser's coin. | <ul style="list-style-type:lower-alpha"><li> Find the probability of HT, TH, HH, TT with two tosses of Prosser's coin. | ||
</li> | </li> | ||
<li> Using part (a), show that the probability of a win on any one play is 1/2, no | <li> Using part (a), show that the probability of a win on any one play is 1/2, no | ||
Latest revision as of 23:06, 12 June 2024
You are playing heads or tails with Prosser but you suspect that his coin is unfair. Von Neumann suggested that you proceed as follows: Toss Prosser's coin twice. If the outcome is HT call the result win. if it is TH call the result lose. If it is TT or HH ignore the outcome and toss Prosser's coin twice again. Keep going until you get either an HT or a TH and call the result win or lose in a single play. Repeat this procedure for each play. Assume that Prosser's coin turns up heads with probability [math]p[/math].
- Find the probability of HT, TH, HH, TT with two tosses of Prosser's coin.
- Using part (a), show that the probability of a win on any one play is 1/2, no matter what [math]p[/math] is.