Question

Calculate the Probability *Binomial Distribution*

An insurance company sells life insurance policies to 5 people with the same age (35) and the same health condition (Excellent). If the Insurance company was estimating at 66% the probability of a person of that age and health condition should live 30 more years. What is the probability that 30 years after buying the policy:

a. All 5 are alive

b. 4 out of 5 have died

c. No more than 3 survive

d. 2 or more survive

Calculate mean, variance and Standard Deviation

Answer 1

Let X be the random variable that denotes the number of people living 30 years after buying the policy.

An insurance company sells life insurance policies to 5 people with the same age and the same health condition. Therefore, n = 5

The probability of a person of that age and health condition living 30 more years is estimated at 66%. Therefore, p = 0.66

X Binomial (n = 5, p = 0.66)

The pmf of X is

P(X = x) = 5Cx * 0.66x * (1 - 0.66)5 - x ; x = 0, 1, 2, 3, 4, 5.

              = 0                                          ; otherwise

a. Answer :

P(X = 5) = 5C5 * 0.665 * (1 - 0.66)5 - 5

              = 0.665

              = 0.1252

Therefore, the probability that 30 years after buying the policy, all 5 are alive is 0.1252

b. Answer :

P(X = 1) = 5C1 * 0.661 * (1 - 0.66)5 - 1

              = 0.0441

Therefore, the probability that 30 years after the buying the policy, 4 out of 5 have died is 0.0441

c. Answer :

P(X 3) = 1 - P(X > 3)

               = 1 - (P(X = 4) + P(X = 5))

               = 1 - (5C4 * 0.664 * (1 - 0.66)5 - 4 + 5C5 * 0.665 * (1 - 0.66)5 - 5)

               = 1 - (0.3226 + 0.1252)

               = 0.5522

Therefore, the probability that 30 years after buying the policy, no more than 3 survive is 0.5522

d. Answer :

P(X 2) = 1 - P(X < 2)

               = 1 - (P(X = 0) + P(X = 1))

               = 1 - (5C0 * 0.660 * (1 - 0.66)5 - 0 + 5C1 * 0.661 * (1 - 0.66)5 - 1)

               = 1 - (0.0045 + 0.0441)

               = 0.9514

Therefore, the probability that 30 years after buying the policy, 2 or more survive is 0.9514

e. Answer :

E(X) = n * p

        = 5 * 0.66

        = 3.3

V(X) = n * p * (1 - p)

        = 5 * 0.66 * (1 - 0.66)

        = 1.122

SD(X) =

           =

           = 1.0593

Therefore, the mean is 3.3, variance is 1.122 and standard deviation is 1.0593