Question

A friend of yours can shoot free throws with 70% accuracy (she makes 70% of her shots). If she attempts 25 free throws, what's the probability she'll make fewer than 14

Answer 1

4.01% probability she'll make fewer than 14

Explanation:

For each free throw, there are only two possible outcomes. Either she makes it, or she does not. Free throws are independent of each other. So the binomial distribution is used. We have a large sample size, which means that we can use the binomial approximation to the normal.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

`E(X) = np`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

Normal probability distribution

Problems of normally distributed distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

She shots 25 free throws, each with a 70% probability of making. So
`n = 25, p = 0.7`

The mean and the standard deviation are given by:

`\mu = E(X) = 25*0.7 = 17.5`

`\sigma = √(V(X)) = √(np(1-p)) = √(25*0.7*0.3) = 2.29`

What's the probability she'll make fewer than 14

Using continuity correction, this is the probability of X being fewer than 14 - 0.5 = 13.5, so this is the pvalue of Z when X = 13.5. Then

`Z = (X - \mu)/(\sigma)`

`Z = (13.5 - 17.5)/(2.29)`

`Z = -1.75`

`Z = -1.75` has a pvalue of 0.0401

4.01% probability she'll make fewer than 14