exercise:42adfb3421: Difference between revisions
(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> Consider the game of tennis when ''deuce'' is reached. If a player wins the next point, he has ''advantage.'' On the following point, he either wins the game or the game returns to ''deuce.'' Assume that for any point, player A has probabilit...") |
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\newcommand{\mathds}{\mathbb}</math></div> Consider the game of tennis when ''deuce'' is | \newcommand{\mathds}{\mathbb}</math></div> Consider the game of tennis when ''deuce'' is reached. If a player wins the next point, he has ''advantage.'' On the following point, he either wins the game or the game returns to ''deuce.'' Assume that for any point, player A has probability .6 of winning the point and player B has probability .4 of winning the point. | ||
reached. If a player wins the next point, he has ''advantage.'' On the | |||
following | <ul style="list-style-type:lower-alpha"><li> Set this up as a Markov chain with state 1: A wins; 2: B wins; 3: | ||
point, he either wins the game or the game returns to ''deuce.'' Assume | |||
that for | |||
any point, player A has probability .6 of winning the point and player B has | |||
probability .4 of winning the point. | |||
<ul><li> Set this up as a Markov chain with state 1: A wins; 2: B wins; 3: | |||
advantage A; 4: deuce; 5: advantage B. | advantage A; 4: deuce; 5: advantage B. | ||
</li> | </li> | ||
| Line 21: | Line 16: | ||
</li> | </li> | ||
</ul> | </ul> | ||
[[exercise:934f183436|Exercise]] and [[exercise:Dcf7521d90|Exercise]] concern the inheritance of color-blindness, which is a sex-linked characteristic. | |||
color-blindness, which is a sex-linked characteristic. | There is a pair of genes, g and G, of which the former tends to produce color-blindness, the latter normal vision. The G gene is dominant. But a man has only one gene, and if this is g, he is color-blind. A man inherits one of his mother's two genes, while a woman inherits one gene from each parent. Thus a man may be of type G or g, while a woman may be type GG or Gg or gg. We will study a process of inbreeding similar to that of [[guide:52e01d4de7#exam 11.1.9 |Example]] by constructing a Markov chain. | ||
There is a | |||
pair of genes, g and G, of which the former tends to produce color-blindness, | |||
the | |||
latter normal vision. The G gene is dominant. But a man has only one gene, | |||
and if this is g, he is color-blind. A man inherits one of his mother's two | |||
genes, while a woman inherits one gene from each parent. Thus a man may be of | |||
type G or g, while a woman may be type GG or Gg or gg. We will study a process | |||
of inbreeding similar to that of [[guide:52e01d4de7#exam 11.1.9 |Example]] by constructing a | |||
Markov chain. | |||
Latest revision as of 23:39, 15 June 2024
Consider the game of tennis when deuce is reached. If a player wins the next point, he has advantage. On the following point, he either wins the game or the game returns to deuce. Assume that for any point, player A has probability .6 of winning the point and player B has probability .4 of winning the point.
- Set this up as a Markov chain with state 1: A wins; 2: B wins; 3: advantage A; 4: deuce; 5: advantage B.
- Find the absorption probabilities.
- At deuce, find the expected duration of the game and the probability that B will win.
Exercise and Exercise concern the inheritance of color-blindness, which is a sex-linked characteristic. There is a pair of genes, g and G, of which the former tends to produce color-blindness, the latter normal vision. The G gene is dominant. But a man has only one gene, and if this is g, he is color-blind. A man inherits one of his mother's two genes, while a woman inherits one gene from each parent. Thus a man may be of type G or g, while a woman may be type GG or Gg or gg. We will study a process of inbreeding similar to that of Example by constructing a Markov chain.