Question

Fredie is a good free throw shooter.Over the last few seasons he has made 80% of his free throws. If fredies shot is independent of the others.At the end of each practice, Fredie shoots 20 free throws to keep sharp.

1. What is the probability that he makes all 20?
2. What is the probability that he makes at least 15?

Answer 1

Let $X=$ the number of sharp free throws: $X\sim Bin(n, p).$

Given $n=20, p=0.8, q=1-p=0.2.$

1.

$$P(X=20)=\dbinom{20}{20}(0.8)^{20}(0.2)^{20-20}=(0.8)^{20}$$

$$\approx0.011529$$

2.

$$P(X\geq 15)=P(X=15)+P(X=16)$$

$$+P(X=17)+P(X=18)$$

$$+P(X=19)+P(X=20)$$

$$=\dbinom{20}{15}(0.8)^{15}(0.2)^{20-15}+\dbinom{20}{16}(0.8)^{16}(0.2)^{20-16}$$

$$+\dbinom{20}{17}(0.8)^{17}(0.2)^{20-17}+\dbinom{20}{18}(0.8)^{18}(0.2)^{20-18}$$

$$+\dbinom{20}{19}(0.8)^{19}(0.2)^{20-19}+\dbinom{20}{20}(0.8)^{20}(0.2)^{20-20}$$

$$\approx0.804208$$