a high school baseball player has a 0.249 batting average. in one game, he gets 7 at bats. what is the…

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, $n = 7$, $p=0.249$, and $1 - p = 0.751$. We want $P(X\geq3)=1-(P(X = 0)+P(X = 1)+P(X = 2))$.

Step2: Calculate $P(X = 0)$

$C(7,0)=\frac{7!}{0!(7 - 0)!}=1$. Then $P(X = 0)=C(7,0)\times(0.249)^{0}\times(0.751)^{7}=1\times1\times(0.751)^{7}\approx0.133$.

Step3: Calculate $P(X = 1)$

$C(7,1)=\frac{7!}{1!(7 - 1)!}=\frac{7!}{1!6!}=7$. Then $P(X = 1)=C(7,1)\times(0.249)^{1}\times(0.751)^{6}=7\times0.249\times(0.751)^{6}\approx7\times0.249\times0.177\approx0.299$.

Step4: Calculate $P(X = 2)$

$C(7,2)=\frac{7!}{2!(7 - 2)!}=\frac{7\times6}{2\times1}=21$. Then $P(X = 2)=C(7,2)\times(0.249)^{2}\times(0.751)^{5}=21\times(0.249)^{2}\times(0.751)^{5}\approx21\times0.062\times0.236\approx0.309$.

Step5: Calculate $P(X\geq3)$

$P(X\geq3)=1-(0.133 + 0.299+0.309)=1 - 0.741 = 0.259$.

Answer:

$0.259$