/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 29 A person bets 1 dollar to \(b\) ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 29

A person bets 1 dollar to \(b\) dollars that he can draw two cards from an ordinary deck of cards without replacement and that they will be of the same suit. Find \(b\) so that the bet is fair.

Short Answer

Expert verified
The value of \(b\) that makes the bet fair is the solution to the equation \(Pb - (1 - P) \times 1 = 0\), where \(P\) is the probability of drawing two cards of the same suit without replacement from a deck of 52 cards. Note that \(P = \frac{13}{52} \times \frac{12}{51}\).

Step by step solution

01

Calculating Probability

Start by finding the probability of drawing two cards of the same suit with no replacement. There are four suits in a deck of 52 cards, so initially, there are 13 cards of the same suit. When one card is drawn, there are now 12 cards of the same suit out of 51 cards total. Therefore, the probability \(P\) is \(\frac{13}{52} \times \frac{12}{51}\). Do the math to find the actual value of \(P\).
02

Finding Value of the Bet

With a bet of 1 dollar to \(b\) dollars, the individual either gains \(b\) dollars or loses 1 dollar. A fair game, by definition, is a game in which the expected payoff is zero. So, create an equation using the known probabilities: \(P \times b - (1 - P) \times 1 = 0\), and solve for \(b\).
03

Solving for \(b\)

Finally, solve the equation to find the value of \(b\) that would make the bet fair. The result is the \(b\) value.
04

Verification

Unpack the equation and verify the result. Make sure the value of \(b\) obtained indeed allows the expected payoff to be zero.

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