Question

Bob is a recent law school graduate who intends to take the state bar exam. According to the National Conference on Bar Examiners, about 55% of all people who take the state bar exam pass. Let n = 1, 2, 3, ... represent the number of times a person takes the bar exam until the first pass.

(a) Write out a formula for the probability distribution of the random variable n. (Use p and n in your answer.)
P(n) =

(b) What is the probability that Bob first passes the bar exam on the second try (n = 2)? (Use 3 decimal places.)


(c) What is the probability that Bob needs three attempts to pass the bar exam? (Use 3 decimal places.)


(d) What is the probability that Bob needs more than three attempts to pass the bar exam? (Use 3 decimal places.)


(e) What is the expected number of attempts at the state bar exam Bob must make for his (first) pass? Hint: Use μfor the geometric distribution and round.

Answer 1

Answer:

Given that:

Bob is a recent law school graduate who intends to take the state bar exam. According to the National Conference on Bar Examiners, about 55% of all people who take the state bar exam pass. Let n = 1, 2, 3, ... represent the number of times a person takes the bar exam until the first pass.

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

Write the formula for the probability distribution of the random variable n

From the given information the random variable n denotes the number of times a person takes the bar exam until the fist pass with the probability of passing the bar exam is 0.55.

Hence, the random variable n follows geometric distribution and its probability mass function is given below:

Hence, the formula for the probability distribution of the random variable n is

(b) What is the probability that Bob first passes the bar exam on the second try (n = 2)   

Find the probability that Bob first passes the bar exam on the second try

The required probability is,

  

  

Thus, the probability that bob first passes the bar exam on the second try is 0.112

(c) What is the probability that Bob needs three attempts to pass the bar exam

Find the probability that Bob needs three attempts to pass the box exam

The required probability is,

  

  

  

Thus, the probability that Bob needs three attempts to pass the bar exam is  0.051

(d) What is the probability that Bob needs more than three attempts to pass the bar exam?

Find the probability that Bob needs more than three attempts to pass the bar exam
The required probability is,

Thus, the probability that Bob needs more than three attempts to pass the ban exam 0.592

(e) What is the expected number of attempts at the state bar exam Bob must make for his (first) pass​​​​​​​

Find the expected number of attempts at the state bar exam Bob must make for Me first pass.
Formula for expected number of attempts is 1/p
Therefore.

Mean = 1/p

= 1/0.55

= 1.818

= 2

Thus. the expected number of attempts at the state bar exam Bob must make for Me fast pass is 2