# Assignment 3: Probability and Discrete & Continuous Random Variables

Problem 1

Given three events , , and   such that  , , . Suppose that events  and  are independent and events  and  are mutually exclusive (disjoint).  Answer the following questions.

1. What is the probability that does not occur, ?

1. What is the probability that both and  occur, P?

1. What is the probability that both and  occur, P?

1. What is the probability that  or  occurs, P?

1. What is the probability that  or  occurs, P?

1. What is the probability that  occurs given that  occurred, P(A| B)?

Problem 2

A major bank in Dubai has Nine employees in the HR department and Seven salespersons (S) in the sales department. The manager of the bank wants to select two employees out of those 16 employees at random to represent the bank at a regional conference.

1. Construct a tree diagram and list all the possible outcomes.

1. What is the probability that the manager selects two salespersons?

1. What is the probability that one HR employee and one salesperson are selected in any order

Problem 3

 A random sample of 235 employees in Dubai was taken. Their University field of study and place of work information were collected and presented in the table below:
 Study Field/Work Place Banking Sector (B) Large scale Companies (C) Wealth Management Firms (W) Total Business (BUS) 65 55 30 150 IT 23 15 12 50 Others (O) 12 15 8 35 Total 100 85 50 235
 a)      What is the probability that a randomly selected employee works in banking sector and his/her field of study is Business?
 P (B Ç BUS) =
 b)      Find the probability that a randomly selected employee works in Banking sector or her/his field of study is Business
 P (B È BUS) =         c)      If an employee works in wealth management firm, what is the probability that her/his field of study is IT?           P (IT ç W) =
1. Are the events IT and W independent

Problem 4:

The members of a firm rent cars from two rental agencies: 60% from agency A, 40% from agency B. If 8% of the cars from agency A need tune-up (T), 10% from B need a tune-up (T). Use Bayes Theorem to answer the questions

1. What is the probability that a rental car delivered to the firm need a tune-up?

1. If a rental car delivered to the firm needs a tune-up, what is the probability that it was from agency B?

Problem 5:

The probability distribution for the rate of return (X) on an investment is

 x 9.5 9.8 10 10.2 10.6 p(x) 0.1 0.2 0.3 0.3 0.1

• What is the probability that the rate of return will be at least 10%?

• Find the expected value (Mean) and the Variance of the rate of return?

Problem 6:

A manufacturing company regularly conducts quality control checks at specified periods on the products it manufactures. Historically, the failure rate for LED light bulbs that the company manufactures is 5%. Suppose a random sample of 15 LED light bulbs is selected and let the random variable X be the number of defected LED light bulbs.

1. What is the probability that none of the LED light bulbs are defective?

P (X = 0) =

1. Find the probability that exactly two LED light bulbs are defective.

P (X = 2) =

1. Compute the probability that at most two LED light bulbs are defective.

P (X 2) =

1. What is the probability that at least three LED light bulbs are defective?

P (X 3) =

1. What are the mean and variance of the number of defective LED light bulbs?

Problem 7:

Toll-Free Telephone Calls

A sales firm receives, on average, 3 calls per hour on its toll-free number. For any given

hour, find the probability that it will receive the following.

1. At most 3 calls
2. 5 or more calls
3. At least 3 calls

Problem 8: Use the Standard Normal Table or Excel to find the required area in the following questions.

1. P (Z <  −2.56)

1. P (Z <  1.47)
2. P (Z >  2.28)
3. P (−1.83 <  Z  <  1.75)

1. P (1.83 <  Z  <  2.75)

Problem 9: In the following questions find the unknown values s and t

1. P (Z < s) = 1515

s =

1. P (Z > t) = 0.8870

t=

Problem 10:

The time spent by a customer to open a new bank account at any bank in Dubai is normally distributed with a mean of 20 minutes and a standard deviation of 8 minutes.

1. What is the probability that a randomly selected customer will spend less than 23 minutes to open a new bank account in Dubai?

1. Find the probability that a randomly selected customer will spend between 15 minutes and 28 minutes to open a ne

1. What is the minimum and maximum time in minutes in the middle 95% of the time spent to open new accounts by customers in Dubai?

1. Find the 75th percentile of the time spent to complete to open new account by customers in Dubai.