Suppose a basketball player, Player A, made 80% of her free throw attempts last season and that she continues to shoot free throws at the same rate. Assume that free throw attempts are independent. Let the random variable X be the number of free throws that player A makes in her next 15 attempts.
a) What is the distribution of X?
b) Find the mean and standard deviation of X.
c) Find the probability that player A makes exactly 10 free throws.
d) Find the probability that player A makes at least 8 free throws.
To find the probability distribution of the random variable X, which represents the number of free throws Player A makes in her next 15 attempts, we can use the binomial distribution.
In this case, the probability of making a free throw is given as 0.8 (80% in decimal form), while the probability of missing a free throw is the complement, which is 1 - 0.8 = 0.2.
The binomial distribution formula is:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
where:
- P(X = k) is the probability of getting exactly k successes in n trials,
- (n choose k) is the binomial coefficient, which represents the number of ways to choose k successes from n trials,
- p is the probability of success in a single trial, and
- (1 - p) is the probability of failure in a single trial.
In this case, n = 15 (the number of attempts), p = 0.8 (the probability of making a free throw), and (1 - p) = 0.2 (the probability of missing a free throw).
Now, we can calculate the probability distribution for X:
P(X = k) = (15 choose k) * 0.8^k * 0.2^(15 - k)
for k = 0, 1, 2, ..., 15.
To find the probabilities for each value of X, you can substitute the values of k into the formula and calculate the result.