Question

Xavier and Yvon have staked $10 each on a colntossing game. Each player tosses the coin in turn. If it lands heads up, the player tossing the coin gets a point; if not, the other player gets a point. The first player to get three points wins the $20. Now suppose the game has to be called off when Xavier has 2 points, Yvon has 1 point, and Xavier is about to toss the coin. What is a fair way to divide the $20 ? 1. These questions refer to the simplified Problem of Points at the beginning of this sketch.

a. Let's call the interrupted-game case described there (X2,Y1), meaning that Xavier has two points and Yvon has one. How many other cases are there? What are they?
b. Analyze the rest of the cases from part (a). (Only two others are "interesting"; you might begin by disposing of the rest quickly.)
c. State your answers for part (b) as fractions or percentages that apply to any amount of money at stake.
d. Suppose the game required 4 points to win. What unfinished case is analogous to the (X2,Y1) case of the 3-point game? Generalize your answer to describe cases that result in a 43-to- 41 division of stakes for an unfinished n-point game.
e. Analyze (X2,Y1) for a 4-point game (with the same stakes). f. Generalize parts (b) and (e) to an n-point game.

Answer 1

To find a fair way to divide the stakes in an interrupted coin-tossing game, one must analyze the probability of each player winning the game from the current scores and make a division based on those probabilities, considering that a fair coin has a 50% chance of landing on either side.

Step-by-step explanation:

The question asks for a fair way to divide the $20 in a coin-tossing game where the game had to be stopped when Xavier had 2 points, Yvon had 1 point, and it was Xavier's turn to toss. To find the fair division, one must consider the probability of each player winning the game from that point onwards.

If Xavier wins the next toss, he wins the game outright. However, if Yvon wins the toss, the game continues. The other cases that need to be considered (though not as 'interesting' because they are more straightforward) are if Xavier had 2 points and Yvon had 0 points, and if Xavier had 1 point and Yvon had 2 points.

To generalize for an n-point game, you would analyze the probability of each player winning from the unfinished case, apply that to the division of stakes, and then express these fractions or percentages as a function of the total stake.

Example for a Fair Coin

Tossing a fair coin is considered a fair method to decide who goes first in a competition because the chances of landing heads or tails are equal, both at 50%. So if you choose heads, you have a 50% chance of the toss going your way.