7 A gambler pays $10 to enter a game in which two coins are tossed. The gambler receives $25, $5 or nothing if two heads, one head or no heads appear, respectively. (a) Determine the expected return per game (ignore the $10 cost of playing). (b) The gambler decides to play until winning the $25 prize. Letting X denote the number of games played until two heads appear (the final game is the Xth game), find the probability function f(x) of X, show that its sum over all possible values of X is one, and find the expected value µ of X. (Note: this is not the expected return on this version of the game. Note too that X here is not quite the same as a geometric random variable, as it was defined in the lecture notes; however, thinking about such random variables will be helpful here).
7 A gambler pays $10 to enter a game in which two coins are tossed. The gambler receives $25, $5 or nothing if two heads, one head or no heads appear, respectively. (a) Determine the expected return per game (ignore the $10 cost of playing). (b) The gambler decides to play until winning the $25 prize. Letting X denote the number of games played until two heads appear (the final game is the Xth game), find the probability function f(x) of X, show that its sum over all possible values of X is one, and find the expected value µ of X. (Note: this is not the expected return on this version of the game. Note too that X here is not quite the same as a geometric random variable, as it was defined in the lecture notes; however, thinking about such random variables will be helpful here).
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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Transcribed Image Text:7 A gambler pays $10 to enter a game in which two coins are tossed. The gambler receives
$25, $5 or nothing if two heads, one head or no heads appear, respectively.
(a) Determine the expected return per game (ignore the $10 cost of playing).
(b) The gambler decides to play until winning the $25 prize. Letting X denote the number of
games played until two heads appear (the final game is the Xth game), find the probability
function f(x) of X, show that its sum over all possible values of X is one, and find the
expected value µ of X.
(Note: this is not the expected return on this version of the game. Note too that X here
is not quite the same as a geometric random variable, as it was defined in the lecture notes;
however, thinking about such random variables will be helpful here).
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