Question

67% of adults age 55 or older want to reach their 100th birthday. You randomly select 8 adults age 55 years or older and ask them if they want to reach their 100th birthday. The random variable represents the number of adults ages 55 or older who want to reach their 100th birthday.

a) What is the probability that exactly 5 of them say they want to reach their 100th birthday?

b) What is the probability that at most 5 of them say they want to reach their 100th birthday?

c) What is the probability that more than 4 adults say they want to reach their 100th birthday?

d) What is the probability that at least 5 of them say they want to reach their 100th birthday?

e) What is the probability that less than 4 adults say that they want to reach their 100th birthday?

f) What is the mean and standard deviation of the binomial variable?

Please show your work!

Answer 1

Let X be the number f adults ages 55 or older who want to reach their 100th birthday in the sample of 8.

Then, X ~ Binomial(n = 8, p = 0.67)

a)

Probability that exactly 5 of them say they want to reach their 100th birthday

= P(X = 5)

= ⁸C₅ * 0.67⁵ * (1 - 0.67)^8-5

= 0.2717089

b)

Probability that at most 5 of them say they want to reach their 100th birthday

= P(X 5) = 1 - P(X > 5)

= 1 - [P(X = 6) + P(X = 7) + P(X = 8)]

= 1 - [⁸C₆ * 0.67⁶ * (1 - 0.67)^8-6 + ⁸C₇ * 0.67⁷ * (1 - 0.67)^8-7 + ⁸C₈ * 0.67⁸ * (1 - 0.67)^8-8 ]

= 1 - (0.27582570 + 0.16000279 + 0.04060677)

= 0.5235647

c)

Probability that more than 4 adults say they want to reach their 100th birthday

= P(X > 4) = P(X 5) = P(X = 5) + P(X > 5)

= ⁸C₅ * 0.67⁵ * (1 - 0.67)^8-5 + (0.27582570 + 0.16000279 + 0.04060677) (From part (b))

= 0.2717089 + 0.4764353

= 0.7481442

d)

Probability that at least 5 of them say they want to reach their 100th birthday

= P(X 5)

= 0.7481442 (From part (c)

e)

Probability that less than 4 adults say that they want to reach their 100th birthday

= P(X < 4)

= 1 - P(X 4)

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

= 1 - [ ⁸C₄ * 0.67⁴ * (1 - 0.67)^8-4 + 0.7481442]

= 1 - (0.1672835 + 0.7481442)

= 0.0845723

f)

Mean = np = 8 * 0.67 = 5.36

Standard deviation = = 1.329962