Question

7. Binomial again

About 40% of the residents aged 25 and older in San Jose CA have completed a college degree.

a. If you were to do a very small marketing research survey of 5 people, what is the chance you would get exactly 2 people with college degrees (assuming everyone was 25 or older)?

 

b. What is the chance you would get at least two people with college degrees in your sample?

(Hint: make a table of all the possible numbers of people out of 5 with college degrees, and calculate the probabilities for the relevant numbers. ).

c. What is the chance you would get at most two people with college degrees in your sample? (same hint as above)

d. How many college graduates do you expect to have in your sample of 5?

Answer 1

Answer:

Given,

p = 0.4

n = 5

Let us consider the binomial distribution i.e,

substitute p , n values

P(X = x) = 5Cx * 0.4^x * (1 - 0.4)^5-x

a)

To determine the required probability

P(X = 2) = 5C2 * 0.4² * 0.6³

= 5 * 0.4^{2 *} 0.6³

P(X = 2) = 0.3456

b)

To determine probability for atleast two people

i.e.,

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

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

= 1 - (5C0 * 0.4⁰ * 0.6⁵ + 5C0 * 0.4¹ * 0.6⁴ )

= 1 - 0.337

= 0.663

Hence our required probability is P(X > 2) = 0.663

c)

To determine for at most two people

i.e.,

P(X < 2) = P(X < 2) + P(X = 2)

= (1 - P(X > 2)) + P(X = 2)

= (1 - 0.663) + 0.3456

= 0.6826

P(X < 2) = 0.6826

d)

To give the expected value

Expected value = n * p

substitute values

Expected value = 5 * 0.4

Expected value = 2