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Asked Mar 12, 2019

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Step 1

**Introduction to binomial distribution:**

Consider a random experiment involving *n* independent trials, such that the outcome of each trial can be classified as either a “success” or a “failure”. The numerical value “1” is assigned to each success and “0” is assigned to each failure.

Moreover, the probability of getting a success in each trial, *p*, remains a constant for all the *n* trials. Denote the probability of failure as *q*. As success and failure are mutually exclusive, *q *= 1 – *p*.

Let the random variable *X* denote the number of successes obtained from the *n* trials. Thus, *X* can take any of the values 0,1,2,…,*n*.

Then, the probability distribution of *X* is a Binomial distribution with parameters (*n*, *p*) and the probability mass function (pmf) of *X*, that is, of a Binomial random variable, is given as:

Step 2

**Normal approximation to binomial distribution:**

It is given that the size of the sample taken from the population of patients is *n* = 100,000. It is known that the patients will be independent of each other, imp-lying independence among each other. As a result, the 100,000 patients may be considered as 100,000 independent trials.

Consider the event of containing a wine as a “success”. It is said that 1% of patients who undergo laser surgery to correct their vision have serious post laser vision problems. That is, Thus, the probability that a patient who undergo laser surgery to correct their vision have serious post laser vision problems, that is, the probability of success in each trial is *p* = 0.01. Then *q* = 1 – 0.01 = 0.99.

Consider *x* as the number of patients who undergo laser surgery to correct their vision have serious post laser vision problems. Then, *x* has a Binomial distribution with parameters (*n* = 100,000, *p* = 0.01).

The mean and variance for the binomial variate is *E*(*x*) = *n ** *p *and *V*(*x*) = *n ***p **(1 – *p*).

Here, the sample size *n *is large.

The mean and variance of the variable *x* is *E*(*x*) = *n ** *p *= 100,000*0.01 = 1,000 and *V*(*x*) = *n ***p **(1 – *p*) = 100,000*0.01*0.99 = 990.

If *x* follows binomial distribution with parameters *np *and *npq, *the binomial variate tends to normal as follows:

Step 3

**Find the probability that fewer than 950 patients experience post laser vision problems:**

Here, the requirement is number of patients who undergo laser surgery to correct their vision have serious post laser vision problems should be le...

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