Question

According to a recent study, 66% of Californians own a car. If you randomly select 8 Californians, what is the probability that: (a) Two of them own a car? (b) At least one of the owns a car? (c) How many are expected to have cars?

Answer 1

Answer: a) 0.01884167 b) 0.00017858 c) 5

Explanation:

Given : The proportion of Californians own a car = 0.66

Sample size : n=8

We assume that each Californian is independent from other.

Let x be the number of Californians own a car.

Then, X
`\sim` Bin (n=8 , p=0.66)

Binomial probability formula =
`P(X=x)=^nC_xp^x(1-p)^x`

, where p=probability of getting success in each trial.

a) The probability that two of them own a car =

`P(X=2)=^8C_2(0.66)^2(1-0.66)^6\\\\=(8!)/(2!6!)(0.66)^2(0.34)^6=0.01884167`

∴ The probability that two of them own a car is 0.01884167.

(b) The probability at least one of the owns a car =

`P(X\geq1)=1-P(X<1)\\\\=1-P(X=0)\\\\=1-^8C_0(0.66)^0(0.34)^8=0.00017858`

∴ The probability at least one of the owns a car is 0.00017858.

(c) The expected number of Californians own a car =
`\mu=np`

`=(8)(0.66)=5.28\approx5`

Hence, the expected number of Californians own a car = 5