# Binomial probability distribution question #2

It is known that 20% of all persons given a certain medication get very drowsy within 2 minutes. Find the probabilities that among 15 persons given the medication, at most 2 will get very drowsy within 2 minutes.

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate binomial probability to find what is the probability that among 15 persons given a certain medication, at most 2 will get very drowsy within 2 minutes, hence you need to use the formula:

`P(r) = C_n^r*p^r*q^(n-r)`

n denotes the number of persons.

p denotes the probability for persons that take a certain medication to  get very drowsy within 2 minutes.

q denotes the probability for persons that  take a certain medication not to get drowsy within 2 minutes.

p+q = 1

r = 2

The problem provides the value of `p = 0.2 (20%), ` hence you may find the value of q such that: `q = 1 - p =gt q = 1 - 0.2 = 0.8`

You may evaluate the binomial probability that among 15 persons given the medication, at most 2 will get very drowsy within 2 minutes such that:

`P(2) = C_15^2*(0.2)^2*(0.8)^(15-2)`

`` Using the factorial formula for binomial coefficient yields:

`C_15^2 = (15!)/(2!*13!)=gtC_15^2 = (13!*14*15)/(1*2*13!)=gt `

`C_15^2 = 7*15 = 105`

`P(2) = 105*0.04*0.32768*0.32768*0.8*0.8`

`P(2) = 105*0.04*0.10737*0.64 = 0.2886`

Hence, evaluating the probability that among 15 persons given a certain medication, at most 2 will get very drowsy within 2 minutes yields P(2) = 28.86%.