Kim and Susan are playing a tennis match where the winner must win 2 sets in order to win the match. For each set, the probability that Kim wins the set is 0.53. The probability of Kim winning the set is not affected by who has won any previous sets. What is the expected value for the number of sets that Kim will win? (Hint: We only care about the number of sets Kim wins during the match, not if she ended up winning the match in the end.)

To find the expected value for the number of sets that Kim will win, we need to consider the probabilities of different outcomes.

Since Kim must win 2 sets to win the match, the possible outcomes can be categorized as follows:

1. Kim wins both sets: The probability of this outcome is the product of the individual probabilities, 0.53 * 0.53 = 0.2809.

2. Kim wins the first set but loses the second: The probability of this outcome is the product of the probability of Kim winning the first set (0.53) and the probability of Susan winning the second set (1 - 0.53 = 0.47). So the probability of this outcome is 0.53 * 0.47 = 0.2491.

3. Kim loses the first set but wins the second: The probability of this outcome is the product of the probability of Kim losing the first set (1 - 0.53 = 0.47) and the probability of Kim winning the second set (0.53). So the probability of this outcome is 0.47 * 0.53 = 0.2491.

4. Kim loses both sets: The probability of this outcome is the product of the individual probabilities, 0.47 * 0.47 = 0.2209.

Now, let's calculate the expected value for the number of sets that Kim will win. We can assign the following values to the outcomes:

- Kim wins both sets: 2 sets won
- Kim wins the first set but loses the second: 1 set won
- Kim loses the first set but wins the second: 1 set won
- Kim loses both sets: 0 sets won

The expected value is calculated by multiplying the value of each outcome by its probability and summing them up. So the expected value is:

(2 * 0.2809) + (1 * 0.2491 * 2) + (0 * 0.2491) + (0 * 0.2209) = 0.5618 + 0.4982 + 0 + 0 = 1.06

Therefore, the expected value for the number of sets that Kim will win is 1.06.

To find the expected value for the number of sets that Kim will win, we can use the concept of expected value.

Let X be the random variable representing the number of sets Kim wins.

Since each set is independent of the others and the probability of winning a set is 0.53, we can model X as a binomial random variable with parameters n and p, where n is the number of sets played and p is the probability of winning a set.

In this case, we are interested in finding the expected value of X, which is denoted as E(X).

The formula for the expected value of a binomial random variable is E(X) = n * p.

Since we want to find the expected value for the number of sets Kim will win, we need to find the expected value of X.

Let's denote the number of sets played as N.

Since the winner must win 2 sets in order to win the match, we have the following possibilities for the number of sets played:

- If the match ends in exactly 2 sets, then N = 2.
- If the match ends in exactly 3 sets, then N = 3.
- If the match ends in exactly 4 sets, then N = 4.
- And so on.

If we assume that the match could potentially continue indefinitely, we can say that N can take any positive integer value.

To find the expected value of X, we need to consider all possible values of N and calculate the expected value for each case.

Let's calculate the expected value of X for each case:

For N = 2:
E(X|N = 2) = 2 * 0.53 = 1.06

For N = 3:
E(X|N = 3) = 3 * 0.53 = 1.59

For N = 4:
E(X|N = 4) = 4 * 0.53 = 2.12

And so on.

To find the overall expected value of X, we need to take into account the probabilities of each case occurring.

Since the probability of Kim winning a set is independent of the number of sets played, we can assume a geometric distribution for the number of sets played with parameter p = 0.53.

The probability mass function of a geometric distribution is given by P(N = k) = (1 - p)^(k-1) * p.

Let's denote the probability of N = k as P(N = k).

For N = 2:
P(N = 2) = (1 - 0.53)^(2-1) * 0.53 = 0.53

For N = 3:
P(N = 3) = (1 - 0.53)^(3-1) * 0.53 = 0.2481

For N = 4:
P(N = 4) = (1 - 0.53)^(4-1) * 0.53 = 0.1166

And so on.

To find the overall expected value of X, we need to sum the expected values for each case multiplied by their respective probabilities:

E(X) = P(N = 2) * E(X|N = 2) + P(N = 3) * E(X|N = 3) + P(N = 4) * E(X|N = 4) + ...

E(X) = 0.53 * 1.06 + 0.2481 * 1.59 + 0.1166 * 2.12 + ...

To find the exact value of E(X), we need to sum the terms of this infinite series.

However, since the probability of winning a set is greater than 0.5, we can expect that the match will eventually end, and the expected value for X will be finite.

Therefore, we can approximate the expected value of X by considering only a few terms of the series.

For example, let's consider the first 10 terms of the series:

E(X) ≈ 0.53 * 1.06 + 0.2481 * 1.59 + 0.1166 * 2.12 + ...

E(X) ≈ 0.5624 + 0.3932879 + 0.2462792 + ...

E(X) ≈ 1.2019661 + ...

E(X) ≈ 1.2019661

Therefore, the expected value for the number of sets that Kim will win is approximately 1.2019661.