(a) The probability that each of them wins one game can be calculated by considering the two possible outcomes: Samantha wins the first game but loses the second, and Samantha loses the first game but wins the second.
The probability of Samantha winning the first game is 0.7 while Claudia winning the second game (given Samantha won the first) is (1 - 0.8) = 0.2. Thus, the probability of the first scenario (Samantha wins the first game and Claudia wins the second) is 0.7 * 0.2 = 0.14.
Likewise, the probability of Samantha losing the first game is (1 - 0.7) = 0.3 and the probability of Samantha winning the second game given she lost the first game is 0.4. Hence, the probability of the second scenario (Claudia wins the first game and Samantha wins the second) is 0.3 * 0.4 = 0.12.
Adding the probabilities of these two scenarios, we find that the probability that each wins one of two games played is 0.14 + 0.12 = 0.26.
(b) To find the probability distribution for the number of games Samantha wins, we need to consider three scenarios - when she wins zero games, one game, or two games.
The probability that she wins no game is found by multiplying the probability of her losing the first and second game. Which is (1 - 0.7) * (1 - 0.4) = 0.18.
We have already calculated the probability that she wins one game in part (a), which is 0.26.
The probability of her winning both games is found by multiplying the probabilities that she wins the first and also wins the second given she wins the first. This is 0.7 * 0.8 = 0.56.
So, the probability distribution for the number of games she wins is:
- 0 wins: 0.18
- 1 win: 0.26
- 2 wins: 0.56
(c) In order to find the mean and standard deviation, we consider the number of wins as a random variable that can take on the values 0, 1, or 2.
The mean or expected value is calculated as 0*0.18 + 1*0.26 + 2*0.56 = 0.676
The variance can be calculated as ((0 - 0.676)²*0.18) + ((1 - 0.676)²*0.26) + ((2 - 0.676)²*0.56) = 0.474
The standard deviation, which is the square root of the variance, is approximately 0.689.
Therefore, the mean number of games that Samantha wins is roughly 0.676 and the standard deviation is approximately 0.689.