exercise:8ae7bbfa06: Difference between revisions
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\newcommand{\mathds}{\mathbb}</math></div> A gambler plays a game in which on each play he wins | \newcommand{\mathds}{\mathbb}</math></div> A gambler plays a game in which on each play he wins one dollar with probability <math>p</math> and loses one dollar with probability <math>q = 1 -p</math>. The ''Gambler's Ruin problem'' is the problem of finding the probability <math>w_x</math> of winning an amount <math>T</math> before losing everything, starting with state <math>x</math>. Show that this problem may be considered to be an absorbing Markov chain with states 0, 1, 2, ..., <math>T</math> with 0 and <math>T</math> absorbing states. Suppose that a gambler has probability <math>p = .48</math> of winning on each play. Suppose, in addition, that the gambler starts with 50 dollars and that <math>T =100</math> dollars. Simulate this game 100 times and see how often the gambler is ruined. This estimates <math>w_{50}</math>. | ||
one dollar with probability <math>p</math> and loses one dollar with probability <math>q = 1 - | |||
p</math>. The ''Gambler's Ruin problem'' is the | |||
problem of | |||
finding the probability <math>w_x</math> of winning an amount <math>T</math> before losing | |||
everything, starting | |||
with state <math>x</math>. Show that this problem may be considered to be an absorbing | |||
Markov chain with states 0, 1, 2, | |||
Suppose that a gambler has probability <math>p = .48</math> of winning on each play. | |||
Suppose, in addition, that the gambler starts with 50 dollars and that <math>T = | |||
100</math> | |||
dollars. Simulate this game 100 times and see how often the gambler is ruined. | |||
This estimates <math>w_{50}</math>. | |||
Latest revision as of 00:02, 16 June 2024
A gambler plays a game in which on each play he wins one dollar with probability [math]p[/math] and loses one dollar with probability [math]q = 1 -p[/math]. The Gambler's Ruin problem is the problem of finding the probability [math]w_x[/math] of winning an amount [math]T[/math] before losing everything, starting with state [math]x[/math]. Show that this problem may be considered to be an absorbing Markov chain with states 0, 1, 2, ..., [math]T[/math] with 0 and [math]T[/math] absorbing states. Suppose that a gambler has probability [math]p = .48[/math] of winning on each play. Suppose, in addition, that the gambler starts with 50 dollars and that [math]T =100[/math] dollars. Simulate this game 100 times and see how often the gambler is ruined. This estimates [math]w_{50}[/math].