exercise:E6fe303a8b: 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> Suppose that a gambler starts with a stake of 0 dollars. <ul><li> Show that the probability that her stake never reaches <math>M</math> before returning to 0 equals <math>1 - p(1 - q_1)</math>. </li> <li> Show that the probability that her stake...") |
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\newcommand{\mathds}{\mathbb}</math></div> Suppose that a gambler starts with a stake of 0 dollars. | \newcommand{\mathds}{\mathbb}</math></div> Suppose that a gambler starts with a stake of 0 dollars. | ||
<ul><li> Show that the probability that her stake never reaches <math>M</math> before returning to 0 equals | <ul style="list-style-type:lower-alpha"><li> Show that the probability that her stake never reaches <math>M</math> before returning to 0 equals | ||
<math>1 - p(1 - q_1)</math>. | <math>1 - p(1 - q_1)</math>. | ||
</li> | </li> | ||
<li> Show that the probability that her stake reaches the value <math>M</math> exactly <math>k</math> times before | <li> Show that the probability that her stake reaches the value <math>M</math> exactly <math>k</math> times before | ||
returning to 0 equals <math>p(1-q_1)(1 - qq_{M-1})^{k-1}(qq_{M-1})</math>. '' Hint'': | returning to 0 equals <math>p(1-q_1)(1 - qq_{M-1})^{k-1}(qq_{M-1})</math>. '' Hint'': Use [[exercise:026effa772 |Exercise]]. | ||
Use | |||
</li> | </li> | ||
</ul> | </ul> | ||
Latest revision as of 01:57, 15 June 2024
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Suppose that a gambler starts with a stake of 0 dollars.
- Show that the probability that her stake never reaches [math]M[/math] before returning to 0 equals [math]1 - p(1 - q_1)[/math].
- Show that the probability that her stake reaches the value [math]M[/math] exactly [math]k[/math] times before returning to 0 equals [math]p(1-q_1)(1 - qq_{M-1})^{k-1}(qq_{M-1})[/math]. Hint: Use Exercise.