Assuming the binomial distribution applies with a sample size of n = 20, find th

Question

Assuming the binomial distribution applies with a sample size of n = 20, find the values below. a. the probability of 5 or more successes if the probability of a success is 0.40 b. the probability of fewer than 4 successes if the probability of a success is 0.45 c. the expected value of the random variable if the probability of success is 0.25 d. the standard deviation of the random variable if the probability of success is 0.25 a. If the probability of a success is 0.40, the probability of 5 or more successes is Round to four decimal places as needed.)

Explanation / Answer

sample size i.e n= 20

We know that in Bionomial distribution P(X=r) = nCrpr(1-p)n-r

a)

given that p=0.40

probability of 5 or more successes = 1 - probability of fewer than 4 successes

= 1 - [P(X=0) + P(X=1) + P(X=2) +P(X=3) + P(X=4)]

= 1 - [20C00.40(1-0.40)20+20C10.41(1-0.40)20-1+20C20.42(1-0.40)20-2+20C30.43(1-0.40)20-3+20C40.44(1-0.40)20-4]

= 0.0510

b)

given that p=0.45

probability of fewer than 4 successes = P(X=0) + P(X=1) + P(X=2) +P(X=3)

= 20C00.450(1-0.45)20+20C10.451(1-0.45)20-1+20C20.452(1-0.45)20-2+20C30.453(1-0.45)20-3

= 0.0049

c)

given that p= 0.25

expected value of bionomial random variable = np = 20*0.25 = 5

d)

given that p= 0.25

standard deviation of the random variable = np(1-p)

= 20*0.25*(1-0.25) = 3.75

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