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Asked Mar 11, 2019

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I am having trouble answering questions number 4 and 5. Thank you

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

**4.**

**Obtain the probability that at least 2 of the first 10 donors have type B blood:**

The size of the sample taken from the population of blood donors is *n* = 10. It is said that it is a random sample, imp-lying independence among each other. As a result, the 10 blood donors may be considered as 10 independent trials.

Consider the event of having a type B blood as a “success”. American Red Cross says that 11% of the U.S. population has Type B blood. Thus, the probability that a donor having type B blood, that is, the probability of success in each trial is *p* = 0.11. Then *q* = 1 – 0.11 = 0.89.

Consider *X* as the number of donors among the 10 sampled donors, who have the type B blood. Then, *X* has a Binomial distribution with parameters (*n* = 10, *p* = 0.11).

The probability that at least 2 of the first 10 donors have type B blood is obtained as **0.3028 **from the calculation given below:

Step 3

**Normal approximation to binomial distribution:**

The pmf for the binomial distribution or the probability of obtaining *x* successes in *n* independent trails of a binomial experiment is,

*P*(*x*) = *nCx* * *px * *(1 – *p*)*n–x*

The mean and standard deviation for the binomial variate is *E*(*x*) = *n ** *p *and *S.D*(*x*) = SQRT(n **p **(1 – *p*)).

It is given that, about 11% of the U.S. population has Type B blood.

The probability of success (U.S. population with Type B blood) = *p *= 11/100 = 0.11.

The probability of failure (U.S. population not with Type B b...

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