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 find the distribution and parameters for N and S, we need to look at the number of successful free throws made by Mark and John. Let's start with Mark.
N: The number of successful free throws made by Mark
N follows a binomial distribution since each attempt is a binary outcome (success or failure) with a fixed probability of success (90%). The parameters for N are the number of trials (2) and the probability of success (0.9).
S: The number of successful free throws made by John
S also follows a binomial distribution with 3 trials and a probability of success of 50% (0.5).
To find the probability that Mark will win the game, we need to find the probability that N > S.
P(N > S) = P(N = 3, S = 0) + P(N = 2, S = 0) + P(N = 2, S = 1)
To calculate these probabilities, we can use the probability mass function (PMF) of the binomial distribution. The PMF gives the probability of obtaining a specific number of successes in a fixed number of trials.
P(N = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- C(n, k) is the number of combinations or ways to choose k successes out of n trials (n Choose k)
- p is the probability of success for each trial
- n is the number of trials
Using this formula, we can calculate the probability of each event and then add them together to find the total probability of Mark winning.
N follows a binomial distribution with parameters n = 2 (number of free throw attempts) and p = 0.9 (probability of making a basket on each attempt).
S follows a binomial distribution with parameters n = 3 (number of free throw attempts) and p = 0.5 (probability of making a basket on each attempt).
To find the probability that Mark wins the game, we need to calculate the probability that N > S.
P(N > S) = P(N = 2, S = 0) + P(N = 2, S = 1) = P(N = 2) * P(S = 0) + P(N = 2) * P(S = 1)
Let's calculate these probabilities step by step:
P(N = 2) = (2 choose 2) * (0.9)^2 * (1-0.9)^(2-2) = 1 * 0.9^2 * 0.1^0 = 0.81
P(S = 0) = (3 choose 0) * (0.5)^0 * (1-0.5)^(3-0) = 1 * 1 * 0.5^3 = 0.125
P(S = 1) = (3 choose 1) * (0.5)^1 * (1-0.5)^(3-1) = 3 * 0.5 * 0.5^2 = 0.375
Therefore,
P(N > S) = 0.81 * 0.125 + 0.81 * 0.375 = 0.081 + 0.30375 = 0.38475
The probability that Mark wins the game is 0.38475, or approximately 38.5%.