Question

Suppose a basketball player has made 231 out of 361 free throws. If the player makes the next 2 free throws, I will pay you $31. Otherwise you pay me $21.

Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.

Answer 1

The expected value of the proposition is of -0.29.

Explanation:

For each free throw, there are only two possible outcomes. Either the player will make it, or he will miss it. The probability of a player making a free throw is independent of any other free throw, which means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

Suppose a basketball player has made 231 out of 361 free throws.

This means that
`p = (231)/(361) = 0.6399`

Probability of the player making the next 2 free throws:

This is P(X = 2) when n = 2. So

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 2) = C_(2,2).(0.6399)^(2).(0.3601)^(0) = 0.4095`

Find the expected value of the proposition:

0.4095 probability of you paying $31(losing $31), which is when the player makes the next 2 free throws.

1 - 0.4059 = 0.5905 probability of you being paid $21(earning $21), which is when the player does not make the next 2 free throws. So

`E = -31*0.4095 + 21*0.5905 = -0.29`

The expected value of the proposition is of -0.29.