**Problem #3: An Interesting Coin Game**Suppose that two people A and B are playing a game with a single coin which has a probability $ p $ of coming up heads and $ q = 1 - p $ of coming up tails. The game begins with A flipping the coin and then B flipping the coin and then A flipping the coin and then B flipping, and so on, until the coin comes up heads. The winner of the game is the one that flips a heads on the coin.a.) The game is in no person’s favor if the probability that A wins is the same as the probability that B wins, both being 1/2. Show that the game is always in A’s favor (i.e., $ P(A) > 1/2 $ and $ P(B) < 1/2 $ for any $ 0 < p < 1 $.b.) Suppose that it costs A $a (which goes to the Casino) to play the game and it cost B $b (which goes to the Casino) to play the game and suppose that the winner of the game gets $c (c > a and c > b) from the Casino. The game is called fair to a given player if the average winnings (per play) for that player is $0. Determine (in terms of $ c $ and $ p $) the values of $ a $ and $ b $ if the game is to be fair to both players.

Name: **Problem #3: An Interesting Coin Game**Suppose that two people A and
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Description: **Problem #3: An Interesting Coin Game**Suppose that two people A and B are playing a game with a single coin which has a probability $ p $ of coming up heads and $ q = 1 - p $ of coming up tails. The game begins with A flipping the coin and the

It is given that there are two-player A and B and they are playing a game where they will toss a coin and the player who will get first a head that player will be the winner otherwise the game will continue until one of the players will not win the game. the probability that the comes Head is p. The game starts with player A. Thus, PHead=pPTail=q=1-pp=12a). The probability that A wins the game, Therefore, The number of cases would be: Number of cases Players Sequence for flipping the coin and win Outcome Sequence Probability with events 1 A H p 2 A, B, A T, T, H q, q, p 3 A, B, A, B, A T, T, T, T, H q, q, q, q, p 4 A, B, A, B, A, B, A T, T, T, T, T, H q, q, q, q, q, q, p 5 A, B, A, B, A, B, A, B, A T, T, T, T, T, T, H q, q, q, q, q, q, q, q, p .. ….. …. ….Thus, the probability that A win is: PA=PWin A=p+qqp+qqqqp+qqqqqqp+.... =p1-q2 =121-14 =23 PA=23>12 Thus, the P(A) is greater than 1/2.Similarly for Player B, The probability that A wins the game, Therefore, The number of cases would be: Number of cases Players Sequence for flipping the coin and win Outcome Sequence Probability with events 1 A, B T, H q, p 2 A, B, A, B T, T, T, H q, q, q, p 3 A, B, A, B, A, B T, T, T, T, T, H q, q, q, q, q, p 4 A, B, A, B, A, B, A, B T, T, T, T, T, T, T, H q, q, q, q, q, q, q, p .. ….. …. ….Thus, the probability that B win is: PB=PWin B=qp+qqqp+qqqqqp+.... =pq1-q2 =12×121-14, because p=PHead=12 =13 =23 Thus, PB=13<12 Hence prooved.b). It is given that the A has $a before playing the game while the B has $b before playing the game. Assume that p is the probability that each player wins the game because the game is fair for both players. Therefore, the probability would be equal for both. It is given that the winning player will get $c. And the average winnings for the player is $0. The formula, for expected money won, is: E=winning moneyPwin the game-Lose the initial moneyPlose the game Therefore, cp-a1-p=0cp-b1-p=0From the above equation, the value of a and b is: a=cp1-pb=cp1-p This is the required answer.

Math

Probability

Problem #3 An Interesting Coin Game Suppose that two people A and B are playing a game with a single coin which has probability p of coming up heads and q = 1 – p of coming up tails. The game begins with A flipping the coin and then B flipping the coin and then A flipping the coin and then B flipping, and so on, until the coin comes up heads. The winner of the game is the one that flips a heads on the coin. а.) uie provaviity that B wins, both being 1/2. Show that the game is always in A’s favor (i.e., P(A) > 1/2 and P(B) < 1/2) for any 0 < p < 1. b.) cost B $6 (which goes to the Casino) to play the game and suppose that the winner of the game gets Sc (c > a and c > b) from the Casino. The game is called fair to a given player if the average winnings (per play) for that player is $0. Determine (in terms of c and p) the values of a and b if the game is to be fair to both players. The game is in no person's favor if the probability that A wins is the same as Suppose that it cost A $a (which goes to the Casino) to play the game and it

Problem #3 An Interesting Coin Game Suppose that two people A and B are playing a game with a single coin which has probability p of coming up heads and q = 1 – p of coming up tails. The game begins with A flipping the coin and then B flipping the coin and then A flipping the coin and then B flipping, and so on, until the coin comes up heads. The winner of the game is the one that flips a heads on the coin. а.) uie provaviity that B wins, both being 1/2. Show that the game is always in A’s favor (i.e., P(A) > 1/2 and P(B) < 1/2) for any 0 < p < 1. b.) cost B $6 (which goes to the Casino) to play the game and suppose that the winner of the game gets Sc (c > a and c > b) from the Casino. The game is called fair to a given player if the average winnings (per play) for that player is $0. Determine (in terms of c and p) the values of a and b if the game is to be fair to both players. The game is in no person's favor if the probability that A wins is the same as Suppose that it cost A $a (which goes to the Casino) to play the game and it

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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Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

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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**Problem #3: An Interesting Coin Game**

Suppose that two people A and B are playing a game with a single coin which has a probability $ p $ of coming up heads and $ q = 1 - p $ of coming up tails. The game begins with A flipping the coin and then B flipping the coin and then A flipping the coin and then B flipping, and so on, until the coin comes up heads. The winner of the game is the one that flips a heads on the coin.

a.) The game is in no person’s favor if the probability that A wins is the same as the probability that B wins, both being 1/2. Show that the game is always in A’s favor (i.e., $ P(A) > 1/2 $ and $ P(B) < 1/2 $ for any $ 0 < p < 1 $.

b.) Suppose that it costs A $a (which goes to the Casino) to play the game and it cost B $b (which goes to the Casino) to play the game and suppose that the winner of the game gets $c (c > a and c > b) from the Casino. The game is called fair to a given player if the average winnings (per play) for that player is $0. Determine (in terms of $ c $ and $ p $) the values of $ a $ and $ b $ if the game is to be fair to both players.

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