Mark and job are basketball players and will play a game. In the game, Mark will get 2 free throw attempts and John will get three. Overall, Mark makes a basket on 90% of free throw attempts and John makes it 50% of the time. Mark will win the game if he makes more free throws than John. If it's a tie then John wins. Assuming independence. Let N denote the number of baskets made by Mark and S denote the number made by John.

What is the distribution and parameters for N and for S? What is the probability that Nash will win the game?

To determine the distribution and parameters for N and S, we need to understand the concept of the binomial distribution.

1. Distribution and Parameters for N (Mark's number of baskets):
Mark has 2 free throw attempts, and he makes a basket on 90% of his attempts. Therefore, the distribution for N follows a binomial distribution with parameters n = 2 (number of trials) and p = 0.9 (probability of success on each trial). We can denote it as N ~ Binomial(2, 0.9).

2. Distribution and Parameters for S (John's number of baskets):
John has 3 free throw attempts, and he makes a basket on 50% of his attempts. Hence, the distribution for S follows a binomial distribution with parameters n = 3 (number of trials) and p = 0.5 (probability of success on each trial). We can denote it as S ~ Binomial(3, 0.5).

Now, let's calculate the probability that Mark will win the game.

To find the probability of Mark winning, we need to consider the possible outcomes and their respective probabilities.

1. Mark wins if N > S.
This occurs when Mark makes more baskets than John or if it's a tie (John makes the same number of baskets as Mark).

2. Calculate the probabilities of each outcome.
To find the probability of each outcome, we need to calculate the probabilities of all possible values of N and S and sum up the probabilities for Mark winning.

P(N > S) = P(N = 0, S = 0) + P(N = 0, S = 1) + P(N = 1, S = 0) + P(N = 1, S = 1) + P(N = 2, S = 1) + P(N = 2, S = 2)

3. Use the binomial distribution formula to calculate each probability.
The binomial distribution formula is P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) represents the binomial coefficient.

P(N = 0, S = 0) = C(2, 0) * 0.9^0 * (1-0.9)^(2-0) * C(3, 0) * 0.5^0 * (1-0.5)^(3-0)
P(N = 0, S = 1) = C(2, 0) * 0.9^0 * (1-0.9)^(2-0) * C(3, 1) * 0.5^1 * (1-0.5)^(3-1)
P(N = 1, S = 0) = C(2, 1) * 0.9^1 * (1-0.9)^(2-1) * C(3, 0) * 0.5^0 * (1-0.5)^(3-0)
P(N = 1, S = 1) = C(2, 1) * 0.9^1 * (1-0.9)^(2-1) * C(3, 1) * 0.5^1 * (1-0.5)^(3-1)
P(N = 2, S = 1) = C(2, 2) * 0.9^2 * (1-0.9)^(2-2) * C(3, 1) * 0.5^1 * (1-0.5)^(3-1)
P(N = 2, S = 2) = C(2, 2) * 0.9^2 * (1-0.9)^(2-2) * C(3, 2) * 0.5^2 * (1-0.5)^(3-2)

4. Calculate the sum of the probabilities for Mark winning.
Add up the probabilities calculated in step 3 to find the total probability of Mark winning.

P(Mark wins) = P(N > S) = P(N = 0, S = 0) + P(N = 0, S = 1) + P(N = 1, S = 0) + P(N = 1, S = 1) + P(N = 2, S = 1) + P(N = 2, S = 2)

By evaluating these expressions, you can determine the probability that Mark will win the game.

The distribution for N, the number of baskets made by Mark, follows a binomial distribution. The parameters for this distribution are the number of trials (2) and the success probability (0.9), since Mark makes a basket 90% of the time.

The distribution for S, the number of baskets made by John, also follows a binomial distribution. The parameters for this distribution are the number of trials (3) and the success probability (0.5), since John makes a basket 50% of the time.

To calculate the probability that Mark will win the game, we need to find the probability that N is greater than S. We can calculate this by summing the probabilities of all possible outcomes where N is greater than S.

P(N > S) = P(N = 0, S = 1) + P(N = 0, S = 2) + P(N = 1, S = 2) + P(N = 0, S = 3) + P(N = 1, S = 3) + P(N = 2, S = 3)

To calculate each individual probability, we can use the binomial probability formula:

P(X = k) = (nCk) * p^k * (1-p)^(n-k)

Where n is the number of trials, k is the number of successes, p is the success probability, and (nCk) represents the binomial coefficient.

Using this formula, we can calculate the individual probabilities for each outcome and then sum them to get the total probability of Mark winning the game.