Problem number 30 states, a certain game is won when one player gets three points ahead. Two players, A and B, are playing the game, and at a particular point, A is one point ahead. Each player has an equal probability of winning each point. What is the probability that A wins the game? So they tell us that A is one point ahead, so we will say the score is 1-0, but in reality it could be 2-1 or even 51-50. So let's let left on our graph represent A winning, and to our right represent A losing. So at the start we have two options. Either A could win or lose. If A wins, they have a score of 2-0. If they lose, they have a score of 1-1. If they win, then they could either win or lose again, and it would be either 3-0 or 2-1. These are all the combinations we actually need to figure out this problem, but now let's figure out the likelihood of each of these events. So since each move, left or right, is equally likely, going left once has a 50% chance of occurring and going right also has a 50% chance of occurring. From here, half the occurrences will go to the left and half will go to the right, each with an immediate probability of 50%. Which means in general, any events two layers down will have a 25% likelihood of occurring. In this case, all the way on the left, player A would win, since they would be three points ahead of player B. And all the way on the right, the score would be tied, so the likelihood of A winning would be 50%. Lastly, if there is one win and one loss, well, that's as if nothing would have happened, and it's as if the score got reverted back to 1-0. So now let's calculate the probability of winning. So first, there is a 1-4 chance of 100% certainly winning, and there's a 1-2 chance of having a 50% chance of winning, which represented mathematically ends up being 1 times 1-4, and 1-2 times 1-2. Finally, for the last one, there is a 1-4 chance of winning at the percentage that we are currently trying to calculate. So represented mathematically, that would be 1-4 P. So now we have a formula, and let's solve for P. Let's multiply it all out, and add like terms. Let's subtract 1-4 P from both sides, and then multiply both sides by 4-3rds to get that P is equal to 2-3rds. So the likelihood is 2-3. So the question asked us, what is the probability that A wins the game? The answer is 2-3. Letter B.

probability

game theory

player A

winning chances

calculation