Susan is taking Western Civilization this semester on a pass/fail basis. The department teaching the course has a history of passing 84% of the students in Western Civilization each term. Let n = 1, 2, 3, ... represent the number of times a student takes Western Civilization until the first passing grade is received. (Assume the trials are independent.)
(a) Write out a formula for the probability distribution of the random variable n. (Use p and n in your answer.)
(b) What is the probability that Susan passes on the first try (n = 1)? (Use 2 decimal places.)
(c) What is the probability that Susan first passes on the second try (n = 2)? (Use 3 decimal places.)
(d) What is the probability that Susan needs three or more tries to pass Western Civilization? (Use 3 decimal places.)
(e) What is the expected number of attempts at Western Civilization Susan must make to have her (first) pass? Hint: Use μ for the geometric distribution and round.

a)

b) P(1)=0.84

c) P(2)=0.134

d) P(n≥3)=0.026

e) The expected number of attemps is 1.19.

Step-by-step explanation:

We use the geometric distribution to model this random variable.

(a) Write out a formula for the probability distribution of the random variable n.

The expression for the probability of "times a student has to take Western Civilization" is:

(b) What is the probability that Susan passes on the first try (n = 1)?

We can calculate this with the previous expression

(c) What is the probability that Susan first passes on the second try (n = 2)?

This probability considers she failed the first try, and passes on the second try.

(d) What is the probability that Susan needs three or more tries to pass Western Civilization? (Use 3 decimal places.)

This is equal to the probability of passing in the first or second try.

(e) What is the expected number of attempts at Western Civilization Susan must make to have her (first) pass?

The expected value for passing Western Civilization is:

The expected number of attemps is 1.19.

the correct is answer is a 9-sided polygon!