Answer:
Probability: 0.9405
E(x) = 10.0705
σ = 0.9225
Explanation:
Using the binomial distribution, we get that the probability is calculated as
[tex]\begin{gathered} P(x)=nCx\cdot p^x\cdot(1-p)^{n-x} \\ \text{ Where nCx = }\frac{n!}{x!(n-x)!} \end{gathered}[/tex]Where n is the number of trials, p is the probability of success, and x is the number of successful trials. In this case, p = 91.55% = 0.9155 and n = 11. Then, the probability is calculated as
[tex]\begin{gathered} P(x)=11Cx\cdot0.9155^x\cdot(1-0.9155)^{11-x} \\ P(x)=11Cx\cdot0.9155^x\cdot0.0845^{11-x} \end{gathered}[/tex]Now, we need to calculate the probability that the player will make 9 or more free throws. This probability is equal to
[tex]\begin{gathered} P(x\ge9)=P(9)+P(10)+P(11) \\ \text{ Where} \\ P(9)=11C9\cdot0.9155^9\cdot0.0845^{11-9}=0.1774 \\ P(10)=11C10\cdot0.9155^^{10}\cdot0.0845^{11-10}=0.3844 \\ P(11)=11C11\cdot0.9155^{11}\cdot0.0845^{11-11}=0.3787 \\ So \\ P(x\ge9)=0.1774+0.3844+0.3787 \\ P(x\ge9)=0.9405 \end{gathered}[/tex]Therefore, the probability is equal to 0.9405
On the other hand, the expected value and the standard deviation are equal to
[tex]\begin{gathered} E(x)=np \\ E(x)=(11)(0.9155) \\ E(x)=10.0705 \\ \\ \sigma=\sqrt{np(1-p)} \\ \sigma=\sqrt{11(0.9155)(1-0.9155)} \\ \sigma=0.9225 \end{gathered}[/tex]Therefore, the answers are
Probability: 0.9405
E(x) = 10.0705
σ = 0.9225
