91Ó°ÊÓ

It is given that a shipment of batteries is accepted if at most two batteries do not meet the specifications. The probability that a battery will not meet specifications is equal to 2%.

Let Xdenote the number of batteries that do not meet specifications.

Success is defined as getting a battery that does not meet specifications.

The probability of success is computed below:

$$\begin{gathered} p=2\% \\ =\frac{2}{100} \\ =0.02 \end{gathered}$$

The probability of failure is computed below:

$$\begin{gathered} q=1-p \\ =1-0.02 \\ =0.08 \end{gathered}$$

The number of trials (n) is equal to 50.

The binomial probability formula used to compute the given probability is as follows:

$P\left(X=x\right)={}^n{C}_x{\left(p\right)}^x{\left(q\right)}^{n-x}$

Using the binomial probability formula, the probability that at most two batteries do not meet specifications is computed below:

$$\begin{gathered} P\left(Xâ\copyright ½2\right)=P\left(X=0\right)+P\left(X=1\right)+P\left(X=2\right) \\ ={}^{50}{C}_0{\left(0.02\right)}^0{\left(0.08\right)}^{50-0}+{}^{50}{C}_1{\left(0.02\right)}^1{\left(0.08\right)}^{50-1}+{}^{50}{C}_2{\left(0.02\right)}^2{\left(0.08\right)}^{50-2} \\ =0.36417+0.371602+0.185801 \\ =0.922 \end{gathered}$$

Thus, the probability that the shipment of batteries will be accepted is equal to 0.922.

The probability suggests that 92% of such shipments will be accepted, and only 8% of the shipments will be rejected.

Since the probability of rejection of shipment is low, the quality of batteries produced is good.