Question

Consider a binomial probability distribution with p=0.65 and n=8. Determine the probabilities below.

a) P(x=2)

b) P(x<=1)

c) P (x >6)

Answer 1

We are given

n = 8, p = 0.65

Formula for binomial probability is given as below:

P(X=x) =nCx*p^x* (1 – p)^(n – x)

Part a

P(X=2) = 8C2*0.65^2*( 1 – 0.65)^(8 – 2)

P(X=2) = 28* 0.4225* 0.001838

P(X=2) = 0.021747

Required probability = 0.021747

Part b

P(X ≤ 1) = P(X=0) + P(X=1)

P(X=0) = 8C0*0.65^0*( 1 – 0.65)^(8 – 0)

P(X=0) = 1*1*0.000225

P(X=0) = 0.000225

P(X=1) = 8C1*0.65^1*( 1 – 0.65)^(8 – 1)

P(X=1) = 8*0.65*0.35^7

P(X=1) = 0.003346

P(X ≤ 1) = P(X=0) + P(X=1)

P(X ≤ 1) = 0.000225 + 0.003346

P(X ≤ 1) = 0.003571

Required probability = 0.003571

Part c

P(X>6) = P(X=7) + P(X=8)

P(X=7) = 8C7*0.65^7*( 1 – 0.65)^(8 – 7)

P(X=7) = 8*0.65^7*0.35^1

P(X=7) = 0.137262

P(X=8) = 8C8*0.65^8*( 1 – 0.65)^(8 – 8)

P(X=8) = 1*0.65^8*0.35^0

P(X=8) = 0.031864

P(X>6) = P(X=7) + P(X=8)

P(X>6) = 0.137262 + 0.031864

P(X>6) = 0.169126

Required probability = 0.169126