Answer to Question #322367 in Statistics and Probability for Madi

Question

1. The probability that Tim will sink a foul shot is 70%. If Tim attempts 10

   foul shots, what is the probability that

a)    he sinks exactly 7 shots

b)   he sinks at least 7 shots

c)    he sinks at most 7 shots

d)   he sinks between 4 and 6 shots, inclusive.

Answer 1

Let $X=$ the number of shots sinked by Tim: $X\sim Bin (n, p).$

Given $n=10, p=0.7, q=1-p=0.3.$

a)

$$P(X=7)=\dbinom{10}{7}(0.7)^{7}(0.3)^{10-7}=0.266827932$$

b)

$$P(X\ge 7)=P(X=7)+P(X=8)+P(X=9)$$

$$+P(X=10)=\dbinom{10}{7}(0.7)^{7}(0.3)^{10-7}$$

$$+\dbinom{10}{8}(0.7)^{8}(0.3)^{10-8}+\dbinom{10}{9}(0.7)^{9}(0.3)^{10-9}$$

$$+\dbinom{10}{10}(0.7)^{10}(0.3)^{10-10}=0.6496107184$$

c)

$$P(X\le 7)=1-P(X=8)-P(X=9)$$

$$-P(X=10)=1-\dbinom{10}{8}(0.7)^{8}(0.3)^{10-8}$$

$$-\dbinom{10}{9}(0.7)^{9}(0.3)^{10-9}-\dbinom{10}{10}(0.7)^{10}(0.3)^{10-10}$$

$$=0.6172172136$$

d)

$$P(4\le X\le 6)=P(X=4)+P(X=5)$$

$$+P(X=6)=\dbinom{10}{4}(0.7)^{4}(0.3)^{10-4}$$

$$+\dbinom{10}{5}(0.7)^{5}(0.3)^{10-5}+\dbinom{10}{6}(0.7)^{6}(0.3)^{10-6}$$

$$=0.3397972032$$