suppose that the probability of a dvd player that was manufactured in a certain factory being defective is 2%. what is the probability that 3 dvd players are defective in a shipment of 13 dvd players from this factory?
a. 0.0019
b. 286
c. 0.0023
d. 0.00176
please select the best answer from the choices provided

Question

Accepted Answer

# Answer:
A. 0.0019

# 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: Assign values to the variables
Here, $n = 13$ (number of DVD players), $k=3$ (number of defective DVD players), $p = 0.02$ (probability of a DVD player being defective), and $1-p=0.98$

## Step3: Calculate the combination $C(13,3)$
$$
\begin{align*}
C(13,3)&=\frac{13!}{3!(13 - 3)!}\
&=\frac{13\times12\times11}{3\times2\times1}\
&=286
\end{align*}
$$

## Step4: Calculate $p^{k}$ and $(1 - p)^{n - k}$
$p^{k}=(0.02)^{3}=0.000008$ and $(1 - p)^{n - k}=(0.98)^{10}\approx0.8170728$

## Step5: Calculate the probability $P(X = 3)$
$$
\begin{align*}
P(X = 3)&=C(13,3)\times p^{3}\times(1 - p)^{10}\
&=286\times0.000008\times0.8170728\
&\approx0.0019
\end{align*}
$$

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QUESTION IMAGE

Question

Explanation:

Step1: Identify the binomial probability formula

Step2: Assign values to the variables

Step3: Calculate the combination \(C(13,3)\)

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

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

Answer: