Question

Listed below is the population by state for the 15 states with the largest population. Also included is whether that state's border touches the Gulf of Mexico, the Atlantic Ocean, or the Pacific Ocean (coastline).

Rank	State	Population	Coastline
1	California	38,802,500	Yes
2	Texas	26,956,958	Yes
3	Florida	19,893,297	Yes
4	New York	19,746,227	Yes
5	Illinois	12,880,580	No
6	Pennsylvania	12,787,209	No
7	Ohio	11,459,163	No
8	Georgia	10,097,343	Yes
9	North Carolina	9,943,964	Yes
10	Michigan	9,909,877	No
11	New Jersey	8,938,175	Yes
12	Virginia	8,326,289	Yes
13	Washington	7,061,530	Yes
14	Massachusetts	6,745,408	Yes
15	Arizona	6,731,484	No

Suppose three states are selected at random. Calculate the probability for the following:

1. None of the states selected have a population of more than 9,000,000. (Round your answer to 3 decimal places.)

2. Exactly one of the selected states has a population of more than 9000000 (Round your answer to 3 decimal places.)

3. At least one of the selected states has a population of more than 9000000 (Round your answer to 3 decimal places.)

Answer 1

Total number of states = 15.

Number of states has a population more than 9000000 = 10

p = P ( States has a population more than 9000000) = 10 /15 = 0.6667

n = number of states randomly selected = 3

The random variable X is defined as

X : Number of states has a population more than 9000000.

X takes value 0,1,2,3.

The random variable X follows binomial distribution with n =3 and p = 0.6667.

X ~ Bin ( n=3, p=0.06667)

1) P ( None of the states has a population more than 9000000) = P(X =0)

P ( None of the states has a population more than 9000000) = 0.0370.

2) P ( Exactly one states has a population more than 9000000) = P (X=1)

P ( Exactly one states has a population more than 9000000) = 0.2221.

3) P ( at least one of the states has a population more than 9000000) = P ( X >=1)

= 1- P(X=0)

(form part(1) P(X=0) = 0.0370.)

= 1 - 0.0370 = 0.9630

P ( at least one of the states has a population more than 9000000) = 0.9630.