Question

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

The fair amount that a person should bet (b), given the scenario, is \(b=\frac{51}{12}\) dollars.

Step-by-step solution

Calculate the Probability of Winning

Firstly, one needs to calculate the probability of the events in question. When the person draws the first card, there's no consideration of the suit, so it doesn't affect the outcome. The second card drawn should be of the same suit as the first. There are now 51 cards left in the deck and 12 cards of the same suit as the first card drawn. Hence, the probability \(P\) of drawing a second card of the same suit is \(P=\frac{12}{51}\)

Calculate the Fair Bet

For a bet to be fair, the expected return from the game should be equal to the bet placed. The expected return is calculated as the product of the probability of winning and the potential gain. In this case, the potential gain would be the amount bet \(b\). Therefore, the fair bet \(b\) would be \(1/P\), which simplifies to \(b=\frac{51}{12}\).