Question

A small University in Florida offers STEM (science, technology, engineering, and mathematics) internships to men in STEM majors at the university. A man must be 20 years old or older to meet the age requirement for the internships. The table below shows the probability distribution of the ages of the men in STEM majors at the university.

Ages 17 18 19 20 21 22 23 or older

Probability 0.005 0.107 0.111 0.252 0.249 0.213 0.063

a. Suppose one man is selected at random from the men in STEM majors at the university. What is the probability that the man selected will not meet the age requirement for the internships? Show your work and label your answer with appropriate probability notation.

The university will select a simple random sample of 10 men in STEM majors to participate in a focus group about the internships.

(b) What is the probability that exactly 2 of the 10 men selected will not meet the age requirement for the internships? Show your work, round your answer to four decimal places, and label your answer with appropriate probability notation.

(c) What is the probability that more than 2 of the 10 men selected will not meet the age requirement for the internships? Show your work, round your answer to four decimal places, and label your answer with appropriate probability notation.

Answer 1

a.

Let X be the age of randomly selected man.

Probability that the man selected will not meet the age requirement for the internships = P(X < 20)

= P(X = 17) + P(X = 18) + P(X = 19)

= 0.005 + 0.107 + 0.111

= 0.223

b.

Let Y be the number of men among 10 who does not meet the age requirement for the internships.

Then Y ~ Binomial (n = 10, p = 0.223)

Using Binomial distribution, probability that exactly 2 of the 10 men selected will not meet the age requirement for the internships = P(Y = 2)

= ¹⁰C₂ * 0.223² * (1 - 0.223)^10-2

= 45 * 0.223² *  0.777⁸

= 0.2973

c.

Using Binomial distribution, probability that more than 2 of the 10 men selected will not meet the age requirement for the internships = P(Y > 2) = 1 - P(Y 2)

= 1 - [P(Y = 0) + P(Y = 1) + P(Y = 2)]

= 1 - [¹⁰C₀ * 0.223⁰ * (1 - 0.223)^10-0 + ¹⁰C₁ * 0.223¹ * (1 - 0.223)^10-1 + ¹⁰C₂ * 0.223² * (1 - 0.223)^10-2]

= 1 - [1 * 0.223⁰ *  0.777¹⁰ + 10 * 0.223¹ *  0.777⁹ + 45 * 0.223² *  0.777⁸ ]

= 1 - 0.6077

= 0.3923