Question

A 2-person game of target shooting is played in which each players takes a shot at a target in a sequential manner. If the player who starts the game hits the target, the player wins right away. Otherwise, the opponent will take a turn. If nobody wins in the first round, players continue to take turns and shoot at the target until one of them wins. a) Suppose that Kayla’s probability of shooting the target is 1 while her opponent’s probability of shooting the target is 2 where 2 =p1. The players agree that Kayla always starts the game because she is the weaker player and they set a friendly stake of $1 where the loser gives the winner $1 for each round he/she loses. Let’s denote by the RV Kayla’s profit after games. Find the mean and the variance of ?

Answer 1

The question appears to ask about the mean and variance of Kayla's profit from a shooting game, where probabilities of hitting the target needs to be clarified. Kayla's profit or loss depends on the sequential outcomes of the shots taken by her and her opponent. However, the provided probabilities are ambiguous and need correction before a precise answer can be calculated.

Step-by-step explanation:

Given that Kayla's probability of shooting the target is 1 and her opponent's probability, denoted as p1, equals 2, we need to clarify these probabilities as they should be fractions or decimals representing the chance of a hit. Assuming we correct these values to be between 0 and 1, and Kayla always starts first, we can denote Kayla's probability of hitting the target as p and her opponent's as q. The game is played in sequential rounds until one player hits the target.

The random variable X represents Kayla's profit after several rounds. If Kayla wins the first round, she gains $1. If she loses and her opponent wins, she loses $1. The game follow rounds until one player wins. Therefore, the expected value or mean of Kayla's profit (μ) will be calculated based on the probability of her hitting the target versus the probability of the opponent doing so. The variance (σ^2) would also be computed based on the probability distribution of her winning and losing.

However, the question as stated contains insufficient data and ambiguities that must be resolved before a precise answer can be given. For more accurate results, the probabilities p and q must sum to 1 (i.e., p + q = 1), which corresponds to the certain outcome that either Kayla or her opponent will hit the target in a round.