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.
a) Binomial distribution
b) mean = np = 15 * .80 = ?
sd = √npq = √(15)(.80)(.20) = ?
Note: q = 1 - p
I'll let you finish the calculation.
c) Use a binomial probability table. n = 15, p = .80, x = 10
d) You can approximate a normal distribution by using z-scores.
Formula:
z = (x - mean)/sd
x = 8, mean & sd calculated from b) above.
Use a z-table to determine the probability using the z-score.
I hope this will help get you started.
a) To determine the distribution of X, we need to use the concept of a binomial distribution. In this case, the random variable X represents the number of successes (made free throws) in a fixed number of independent trials (15 attempts), given a constant probability of success (80% free throw percentage) and assuming the attempts are independent.
b) The mean (expected value) of a binomial distribution is calculated by multiplying the number of trials (15) by the probability of success (80% or 0.8).
Mean (μ) = n * p = 15 * 0.8 = 12.
The standard deviation (σ) of a binomial distribution is calculated using the formula:
Standard Deviation (σ) = √(n * p * (1 - p)),
where n is the number of trials and p is the probability of success.
Standard Deviation (σ) = √(15 * 0.8 * (1 - 0.8)) = √(15 * 0.8 * 0.2) = √(2.4) ≈ 1.549.
c) To find the probability that player A makes exactly 10 free throws, we need to use the probability mass function (PMF) formula for the binomial distribution, which is given by:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k),
where (n choose k) is the binomial coefficient, n is the number of trials, p is the probability of success, and k is the number of successes.
In this case, n = 15, p = 0.8, and k = 10.
P(X = 10) = (15 choose 10) * 0.8^10 * (1 - 0.8)^(15 - 10).
To calculate this probability, use the formula for binomial coefficient:
(15 choose 10) = 15! / (10! * (15-10)!) = 3003.
P(X = 10) = 3003 * 0.8^10 * 0.2^5 ≈ 0.2326.
d) To find the probability that player A makes at least 8 free throws, we need to calculate the cumulative probability from 8 to 15 (inclusive) using the binomial distribution.
P(X ≥ 8) = P(X = 8) + P(X = 9) + ... + P(X = 15).
To calculate this probability, you can either add up the individual probabilities for each value of X from 8 to 15, or use a statistical software/tool for binomial distribution calculations.
You can use a binomial distribution calculator or software like R, Python, or Excel to find the cumulative probability directly, which would give you the result: P(X ≥ 8) ≈ 0.9998.