if mike piazza has a batting average of 0.348, what is the probability that he will hit safely 3 times out of the next 10 times at bat?
0.215
0.283
0.253
0.020

Question

Accepted Answer

# 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 $C(n,k)=\frac{n!}{k!(n - k)!}$

## Step2: Determine the values of $n$, $k$, and $p$
Here, $n = 10$ (times at bat), $k = 3$ (hits), and $p=0.348$. Then $1-p = 1 - 0.348=0.652$

## Step3: Calculate the combination $C(n,k)$
$$
\begin{align*}
C(10,3)&=\frac{10!}{3!(10 - 3)!}\
&=\frac{10!}{3!7!}\
&=\frac{10\times9\times8}{3\times2\times1}\
& = 120
\end{align*}
$$

## Step4: Calculate $p^{k}$ and $(1 - p)^{n - k}$
$p^{k}=(0.348)^{3}\approx0.042$ and $(1 - p)^{n - k}=(0.652)^{7}\approx0.041$

## Step5: Calculate the probability $P(X = 3)$
$$
\begin{align*}
P(X = 3)&=C(10,3)\times p^{3}\times(1 - p)^{7}\
&=120\times0.042\times0.041\
&\approx0.215
\end{align*}
$$

# Answer:
0.215

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QUESTION IMAGE

Question

Explanation:

Step1: Identify the binomial probability formula

Step2: Determine the values of \(n\), \(k\), and \(p\)

Step3: Calculate the combination \(C(n,k)\)

Step4: Calculate \(p^{k}\) and \((1 - p)^{n - k}\)

Step5: Calculate the probability \(P(X = 3)\)

Answer: