Question

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?

Short answer

Approximate probability that fewer patients will experience serious post laser vision problem is 0.0537

Step-by-step solution

Given information

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

Calculating the value of mean

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.

Calculating the value of standard deviation

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

Hence the value of standard deviation is 31.646

Checking whether normal distribution is approximate or not.

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

Calculating the value of  P(x<950)

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