3. (4 points) Provide an example scenario of a binomial random variable that is related to your area of work or interest (first, you need to describe your work or area of interest!) and explain clearly how each requirement for the binomial distribution is satisfied. You can use the following questions as guides to provide your example. - Describe briefly your work or area of interest. - Describe a situation related to your work or area of interest with a binomial random variable. Clearly state what the random variable is and what values it can take. - Explain clearly how each of the requirements for the binomial distribution is satisfied in your example.

Using the medical sector scenario of a binomial random variable for example if 90% of adults with allergies report symptomatic relief with a specific medication. And we give the medication to 10 new patients with allergies, what is the probability that it is effective in exactly seven?

Solution

The 'success' = the outcome

The probability of success for each person = 0.9 (90/100)

The final assumption is that the replications are independent, and it is reasonable to assume that this is true.

Number of observations (adults observed) is n=10

Number of successes or events of interest is x=7

p=0.90

The probability of 7 successes is:

[tex]P (X "sucesses")=\frac{10!}{7!(10-7)!}0.90^{7}(1-0.90)^(10-7)}[/tex]

120 × 0.0004782969 = 0.057395628 or 5.7%

Which clearly implies there is a 5.7% probability that exactly 7 of 10 patients will report relief from symptoms when the probability that any one reports relief is 90%.