Question

In the United States, voters who are neither Democrat nor Republican are called Independent. It is believed that 10% of voters are Independent. A survey asked 22 people to identify themselves as Democrat, Republican, or Independent.

A. What is the probability that none of the people are Independent? Probability =

B. What is the probability that fewer than 5 are Independent? Probability =

C. What is the probability that more than 17 people are Independent? Probability =

Answer 1

Solution

Given that ,

p = 0.10

1 - p = 0.90

n = 22

Using binomial probability formula ,

P(X = x) = ((n! / x! (n - x)!) * p^x * (1 - p)^{n - x}

a)

P(X = 0) = ((22! / 0! (22-0)!) * 0.10⁰ * (0.90)^{22 - 0}

= 0.0985

Probability = 0.0985

b)

P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

= ((22! / 0! (22-0)!) * 0.10⁰ * (0.90)^{22 - 0} + ((22! / 1! (22-1)!) * 0.10¹ * (0.90)^{22 - 1} + ((22! / 2! (22-2)!) * 0.10² * (0.90)^{22 - 2} + ((22! / 3! (22-3)!) * 0.10³ * (0.90)^{22 - 3} + ((22! / 4! (22-4)!) * 0.10⁴ * (0.90)^{22 - 4}

= 0.0985 + 0.2407 + 0.2808 + 0.208 + 0.1098

= 0.9379

Probability = 0.9379

c)

P(X > 17) = ( P(X = 18) + P(X = 19) + P(X = 20) + P(X = 21) + P(X = 22)

= (  ((22! / 18! (22-18)!) * 0.10¹⁸ * (0.90)^{22 -18} + ((22! / 19! (22-19)!) * 0.10¹⁹ * (0.90)^{22 - 19} + ((22! / 20! (22-20)!) * 0.10²⁰ * (0.90)^{22 - 20} + ((22! / 21! (22-21)!) * 0.10²¹ * (0.90)^{22 - 21} + ((22! / 22! (22-22)!) * 0.10²² * (0.90)^{22 - 22})

= ( 0 + 0 + 0 + 0 + 0 )

= 0

Probability = 0.000