it is known that a certain lacrosse goalie will successfully make a save 89.94% of the time. suppose that…

Explanation:

Step1: Identify the binomial probability formula

The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single - trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. Here, $n = 10$, $p=0.8994$, and $1 - p = 0.1006$. We want to find $P(X\geq7)=P(X = 7)+P(X = 8)+P(X = 9)+P(X = 10)$.

Step2: Calculate $P(X = 7)$

$C(10,7)=\frac{10!}{7!(10 - 7)!}=\frac{10\times9\times8}{3\times2\times1}=120$. $P(X = 7)=C(10,7)\times(0.8994)^{7}\times(0.1006)^{3}$ $P(X = 7)=120\times(0.8994)^{7}\times(0.1006)^{3}\approx120\times0.4709\times0.001012\approx0.057$

Step3: Calculate $P(X = 8)$

$C(10,8)=\frac{10!}{8!(10 - 8)!}=\frac{10\times9}{2\times1}=45$. $P(X = 8)=C(10,8)\times(0.8994)^{8}\times(0.1006)^{2}$ $P(X = 8)=45\times(0.8994)^{8}\times(0.1006)^{2}\approx45\times0.4236\times0.01012\approx0.193$

Step4: Calculate $P(X = 9)$

$C(10,9)=\frac{10!}{9!(10 - 9)!}=10$. $P(X = 9)=C(10,9)\times(0.8994)^{9}\times(0.1006)^{1}$ $P(X = 9)=10\times(0.8994)^{9}\times0.1006\approx10\times0.3809\times0.1006\approx0.383$

Step5: Calculate $P(X = 10)$

$C(10,10)=\frac{10!}{10!(10 - 10)!}=1$. $P(X = 10)=(0.8994)^{10}\approx0.342$

Step6: Calculate $P(X\geq7)$

$P(X\geq7)=P(X = 7)+P(X = 8)+P(X = 9)+P(X = 10)$ $P(X\geq7)\approx0.057 + 0.193+0.383 + 0.342=0.975$

Answer:

$0.975$