gloria is a softball player. she has a 0.32 chance of hitting the ball each time she bats. if gloria bats 4 times in a game, what is the probability of her getting less than three hits? 0.92 0.02 0.99 0.88

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Accepted Answer

# Explanation:
## Step1: Use 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, $n = 4$, $p=0.32$, and $1-p = 0.68$

The probability of getting less than three hits is $P(X\lt3)=1-(P(X = 3)+P(X = 4))$

## Step2: Calculate $P(X = 3)$
For $k = 3$:
$$
\begin{align*}
C(4,3)&=\frac{4!}{3!(4 - 3)!}=\frac{4!}{3!1!}=\frac{4\times3!}{3!}=4\
P(X = 3)&=C(4,3)\times(0.32)^{3}\times(0.68)^{4 - 3}\
&=4\times0.32^{3}\times0.68\
&=4\times0.032768\times0.68\
&=4\times0.02228224\
&=0.08912896
\end{align*}
$$

## Step3: Calculate $P(X = 4)$
For $k = 4$:
$$
\begin{align*}
C(4,4)&=\frac{4!}{4!(4 - 4)!}=1\
P(X = 4)&=C(4,4)\times(0.32)^{4}\times(0.68)^{4 - 4}\
&=1\times0.32^{4}\times1\
&=0.01048576
\end{align*}
$$

## Step4: Calculate $P(X\lt3)$
$$
\begin{align*}
P(X\lt3)&=1-(P(X = 3)+P(X = 4))\
&=1-(0.08912896 + 0.01048576)\
&=1-0.09961472\
&=0.90038528\approx0.92
\end{align*}
$$

# Answer:
0.92

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QUESTION IMAGE

Question

Explanation:

Step1: Use the binomial probability formula

Step2: Calculate \(P(X = 3)\)

Step3: Calculate \(P(X = 4)\)

Step4: Calculate \(P(X\lt3)\)

Answer: