LASIK surgery complications. According to studies, 1% of all patients who undergo laser surgery (i.e., LASIK) to correct their vision have serious post laser vision problems (All About Vision, 2012). In a sample of 100,000 patients, what is the approximate probability that fewer than 950 will experience serious post laser vision problems?

1% of the patients who undergo laser surgery to correct the vision have serious postlaser vision problem.

Let x be the number of patients undergo the laser surgery to correct the vision who have the serious post laser problem.

With n =100,000 and p =0.001

Hence,

$$\begin{gathered} E\left(x\right)=\mu \\ =np \\ =100,000\times 0.01 \\ E\left(x\right)=1000 \end{gathered}$$

Hence mean is 1000.

$\begin{array}{l}\alpha =\sqrt{npq}\\ =\sqrt{100,000\times 0.01-0.01)}\\ =\sqrt{1000,000\times 0.01\times 0.99}\\ \sigma =31.646\end{array}$

Hence the value of standard deviation is 31.646

Most of the observations will fall within three standard deviation of the mean.

That is, atleast $\frac{8}{9}$of the measurement will fall within 3 standard deviation of the mean.

Lets $\mu =1000$and $\mu =31.464$

The observations will fall within three standard deviation of the mean is,

$\mu \pm 3\sigma =1000\pm 3\left(31.464\right)$

$$\begin{gathered} =\left(1000-3\left(31.464\right),1000+3\left(31.464\right)\right) \\ =\left(1000-94.392,1000+3\left(31.464\right)\right) \end{gathered}$$

$\mu \pm 3\sigma =\left(905,608,1,094.392\right)$

Hence, the normal approximation is appropriate because the interval$\left(905,608,1,094.392\right)$ is lies in the range of 0 to 1000,000

The approximate probability can be obtained by using the formula

$z=\frac{\left(k-0.5\right)-\mu }{\sigma }$

Therefore,

$$\begin{gathered} \mathrm{P}\left(\mathrm{x}<950\right)=\mathrm{P}\left(\mathrm{z}<\frac{\left(950-0.5\right)-\mathrm{\mu }}{\mathrm{\sigma }}\right) \\ =\mathrm{P}\left(\mathrm{z}<\frac{\left(949.5-1000\right)}{31.464}\right) \\ \mathrm{P}\left(\mathrm{x}<950\right)=\mathrm{P}\left(\mathrm{z}<\frac{\left(949.5-1000\right)}{31.464}\right) \\ =\mathrm{P}\left(\mathrm{z}<\frac{-50.5}{31.464}\right) \end{gathered}$$

$P\left(z\le -1.61\right)$[from table 2-normal curve areas]

$\begin{array}{l}=0.5-0.4463\\ =0.0537\end{array}$

Hence, approximate probability that fewer than 950 patients will experience serious post laser vision problem is 0.0537