Answer:
The Probability that she scores at least 8 goals is 52.56%
Step-by-step explanation:
Consider the provided information.
She scores 75% of the time (p) = 0.75
q = 1-p = 1-0.75 = 0.25
She attempts 10 penalty kicks.
Thus, n = 10 , p = 0.75 and q = 0.25
According to the binomial distribution.
[tex]P(X) = ^nC_x p^x q^{n-x}[/tex]
We need to calculate the probability that she scores at least eight goals.
This can be written as:[tex]P(X \geq 8)[/tex]
[tex]P( X\geq 8) = P( X = 8) + P( X = 9) + P( X = 10)[/tex]
[tex]p(X\geq 8)= ^{10}C_8 (0.75)^8(0.25)^2 +^{10}C_9 (0.75)^9 (0.25)+^{10}C_{10} (0.75)^{10}0.25^0[/tex]
[tex]p(X\geq 8)=\frac{10!}{8!2!}(0.75)^8(0.25)^2 +\frac{10!}{9!1!}(0.75)^9 (0.25)+(0.75)^{10}0.25^0[/tex]
[tex]p(X\geq 8)=45(0.75)^8(0.25)^2 +10(0.75)^9 (0.25)+(0.75)^{10}[/tex]
[tex]p(X\geq 8)\approx0.5256=52.56\%[/tex]
Hence, the Probability that she scores at least 8 goals is 52.56%
The probability of scoring exactly 8 times out of 10 is 0.28
We know that:
The general formula that we will use in these cases is:
[tex]P(x) = C(10, x)*P^x*Q^{10 - x}[/tex]
This is the probability of scoring x times out of x.
Where:
[tex]C(N, K) = \frac{N!}{(N - K)!*K!}[/tex]
Now we can just replace x by 8 in the general formula to get:
[tex]P(8) = C(10, 8)*0.75^8*0.25^2 = \frac{10!}{(10 - 8)!*8!} 0.75^8*0.25^2\\\\P(8) = \frac{10*9}{2}*0.75^8*0.25^2 = 0.28[/tex]
So the probability of scoring exactly 8 times out of 10 is 0.28
If you want to learn more about probability, you can read:
https://brainly.com/question/251701
