Question

A game of Plinko is to be played on the board shown below. The pegs are biased such that a disc is twice as likely to move to the right than the left at each peg. Furthermore, the disc can only be dropped from directly above the top-most peg. Answer the following questions about this Plinko game. Be sure to show your work. (a) Let Y be the random variable that describes the winnings when a single dise is dropped. Write down the probability mass function for Y. (b) What is the probability that you win a total of exactly $100 in a game with 2 discs? (c) What is the probability that you win a total of exactly 8200 in a game with 5 dissa? (d) What is the expected winnings if you play a game with 3 dises?

Answer 1

The probability mass function for Y is P(Y = -1) = 1/2 and P(Y = +1) = 1/2. The probability of winning exactly $100 in a game with 2 discs is 2 * (1/2)^51. The probability of winning exactly $8200 in a game with 5 discs is 2 * (1/2)^4110. The expected winnings if you play a game with 3 discs is 0.

Step-by-step explanation:

To find the probability mass function for Y, we first need to determine the possible values that Y can take on. In this game, the disc can move either to the left or the right at each peg, and it is twice as likely to move to the right. The winnings on each side of the board are -1 or +1, respectively. So, the possible winnings when a single disc is dropped are -1 or +1. Since the disc can only be dropped from directly above the top-most peg, the probability of winning -1 or +1 is 1/2 each, as the disc has an equal chance of moving to the left or right at each peg. Therefore, the probability mass function for Y is:

P(Y = -1) = 1/2

P(Y = +1) = 1/2

To find the probability of winning exactly $100 in a game with 2 discs, we need to determine the number of ways in which the discs can land to result in a total winning of $100. Since the disc can move either to the left or the right at each peg, and it is twice as likely to move to the right, the number of ways to achieve a total winning of $100 is the same as the number of ways to get 50 right moves and 1 left move, or 1 right move and 50 left moves. Each combination of moves has a probability of (1/2)^51, as there are 51 moves in total. Therefore, the probability of winning exactly $100 in a game with 2 discs is 2 * (1/2)^51.

To find the probability of winning exactly $8200 in a game with 5 discs, we follow a similar approach as before. The number of ways to achieve a total winning of $8200 is the same as the number of ways to get 4100 right moves and 5 left moves, or 5 right moves and 4100 left moves. Each combination of moves has a probability of (1/2)^4110, as there are 4110 moves in total. Therefore, the probability of winning exactly $8200 in a game with 5 discs is 2 * (1/2)^4110.

To find the expected winnings if you play a game with 3 discs, we need to calculate the expected value of Y. The expected value is given by the sum of each possible outcome multiplied by its probability. In this case, the possible outcomes are -1 and +1, each with a probability of 1/2. Therefore, the expected winnings can be calculated as:

Expected winnings = (-1)(1/2) + (+1)(1/2) = 0