1 Business 260: Managerial Decision Analysis Professor David Mease Lecture 2 Agenda: 1) Assign Homework #1 (due Thursday 3/19) 2) Basic Probability (Stats Book P. 165) 3) Some Important Discrete Probability Distributions (Stats Book P. 195)

2 Homework #1 Homework #1 will be due Thursday 3/19 We will have an exam that day after we review the solutions The homework is posted on the class web page: The solutions are also posted so you can check your answers:

3 Basic Probability (Stats Book P. 165) Statistics for Managers Using Microsoft ® Excel 4 th Edition

4 Chapter Goals After completing this chapter, you should be able to: Explain basic probability concepts and definitions Use contingency tables to view a sample space Apply common rules of probability Compute conditional probabilities Determine whether events are statistically independent

5 In class exercise #21: Figure out the following probabilities. 1) What is the probability of getting a tail on a coin flip? 2) What is the probability of getting two tails in two flips? 3) What is the probability of rolling a 4 with a die? From a single draw from a deck of 52 cards, what is the probability of getting 4) a king? 5) a heart? 6) a red card? 7) a heart if I already know it is a red card? 8) a card that is red AND a king? 9) a card that is red OR a king? 10) a card that is an ace AND a king? 11) If there is a 60% chance of rain, what is the probability it will not rain? 12) In a group of 25 people, what is the probability at least two have the same birthday? (Just guess)

6 Important Probability Rules Mutually Exclusive: P(not A) = 1- P(A) Complement Rule: P(A and B) = 0 P(A or B) = P(A) + P(B) - P(A and B) General Addition Rule (p. 171):

7 Important Probability Rules Conditional Probability (p.176): General Multiplication Rule (p. 180): Statistically Independent (p. 179):

8 In class exercise #22: Pool balls are numbered 1 through 15. Numbers 9 through 15 are striped. Make a contingency table showing striped and solid versus odd and even.

9 In class exercise #23: Using your contingency table answer the following. If I draw a ball at random, what is the probability it is 1) striped 2) striped AND even 3) striped OR even 4) even if I already know it is striped

10 In class exercise #24: Convert your contingency table to a joint probability table. Use this to answer these (same) questions. If I draw a ball at random, what is the probability it is 1) striped 2) striped AND even 3) striped OR even 4) even if I already know it is striped

11 In class exercise #25: Are “striped” and “even” statistically independent?

12 In class exercise #26: A box contains 6 red marbles and 4 green marbles. I randomly draw two with replacement. What is the probability I get 1) a red the first time and a red the second time 2) a green the first time and a red the second time

13 In class exercise #27: A box contains 6 red marbles and 4 green marbles. I randomly draw two with replacement. Is getting a red ball on the second draw independent of getting a red ball on the first draw?

14 In class exercise #28: A box contains 6 red marbles and 4 green marbles. I randomly draw two without replacement. What is the probability I get 1) a red the first time and a red the second time 2) a green the first time and a red the second time

15 In class exercise #29: A box contains 6 red marbles and 4 green marbles. I randomly draw two without replacement. What is the probability I get a red the second time?

16 In class exercise #30: On Friday my friend said that there was a 50 percent chance he would go snowboarding over the weekend. He said that if he goes snowboarding there is about a 40% chance he will break his leg; whereas, if he does not go snowboarding there is only a 2% chance he will break his leg doing something else. 1) What is the chance he breaks his leg over the weekend? 2) If I see him on Monday and his leg is broken, what is the probability he went snowboarding over the weekend? (*NOTE: this is like 31 and 34 on the homework*)

17 In class exercise #31: The probability that a person has a certain disease is A test is available to determine whether a person has the disease. If the person does actually have the disease, the test will indicate so with probability If the person does not have the disease, the test will still indicate the person has the disease with probability Suppose the test says you have the disease. What is the probability that you actually do? (*NOTE: this is like 31 and 34 on the homework*)

18 Some Important Discrete Probability Distributions (Stats Book P. 195) Statistics for Managers Using Microsoft ® Excel 4 th Edition

19 Chapter Goals After completing this chapter, you should be able to: Compute and interpret the mean and standard deviation for a discrete probability distribution Explain covariance and its application in finance Use the binomial probability distribution to find probabilities Describe when to apply the binomial distribution

20 Introduction to Probability Distributions Random Variable Represents a possible numerical value from an uncertain event Random Variables Discrete Random Variable Continuous Random Variable Ch. 5Ch. 6

21 Discrete Probability Distributions A discrete probability distribution is given by a table listing all possible values for the random variable along with the corresponding probabilities. The appropriate chart to display it is a bar chart (which has gaps, unlike a histogram).

22 In class exercise #32: A fair coin is tossed two times. Give the probability distribution and bar chart for the number of tails.

23 In class exercise #33: A fair coin is tossed three times. Give the probability distribution and bar chart for the number of tails.

24 In class exercise #34: A fair die is rolled once. Give the probability distribution and bar chart for the outcome.

25 In class exercise #35: A fair die is rolled twice. Give the probability distribution and bar chart for the total from the two rolls.

26 In class exercise #36: A fair die is rolled twice. Using your probability distribution from before answer the following: A) What is the probability that a seven is rolled? B) What is the probability that the roll is larger than 10? C) What is the probability that an even number is rolled? D) Given the roll is even, what is the probability it is a four? E) What is the probability the roll is even and four? F) What is the probability the roll is four or odd?

27 In class exercise #37: Many people toss a fair coin two times each. How many tails would you expect for each person on average?

28 Discrete Random Variable Summary Measures Expected Value (or mean) of a discrete distribution (Weighted Average)

29 In class exercise #38: A box contains two $1 bills, one $5 bill and one $20 bill. You reach in without looking and pull out a single bill. Give the probability distribution and bar chart for the amount of money you pull out.

30 In class exercise #39: A box contains two $1 bills, one $5 bill and one $20 bill. Many people reach in without looking and each pull out a single bill and put it back. On average, how much money would you expect each person to get? How much money would you personally be willing to pay to play this game once?

31 In class exercise #40: A fair coin is to be tossed two times. A) Give the expected number of tails. B) Give the variance for the number of tails. C) Give the standard deviation for the number of tails.

32 Variance of a discrete random variable Standard Deviation of a discrete random variable where: E(X) = Expected value of the discrete random variable X X i = the ith outcome of X P(X i ) = Probability of the ith occurrence of X Discrete Random Variable Summary Measures (continued)

33 In class exercise #41: Shares of stock X and stock Y each cost $100 dollars per share. Your advisor estimates there is a 20% probability that in one year a share of stock X will be worth $90 and a share of stock Y will be worth $130, a 40% probability X will be worth $100 and Y will be worth $100, and 40% probability X will be worth $130 and Y will be worth $85. Compare the following three investment options in terms of mean, variance and standard deviation. 1) One share of X 2) One share of Y 3) One share of each

34 The Covariance The covariance measures the strength and direction of the linear relationship between two variables I will not ask you to compute it, but here is the formula where:X = discrete variable X X i = the i th outcome of X Y = discrete variable Y Y i = the i th outcome of Y P(X i Y i ) = probability

35 In class exercise #42: Would you guess that the covariance between stock X and stock Y to be positive or negative. Why?

36 The Sum of Two Random Variables Expected Value of the sum of two random variables: Variance of the sum of two random variables: Standard deviation of the sum of two random variables: E(X+Y) = E(X) + E(Y)

37 In class exercise #43: The covariance between stock X and stock Y is Use this information and the formulas on the previous slide to check your answer for the expected value and variance when you buy one share of stock X and one share of stock Y.

38 In class exercise #44: A Bay Area software company is trying to hire as many qualified job candidates as possible. Next Monday they will interview 4 candidates. If the probability of each candidate being hired is 20%, give the probability distribution for the number of candidates they will hire that day assuming that the candidates are independent. Also compute the mean, variance and standard deviation for the number of candidates that will be hired that day.

39 Binomial Probability Distribution A fixed number of observations, n e.g., 15 tosses of a coin; ten light bulbs taken from a warehouse Two mutually exclusive and collectively exhaustive categories e.g., head or tail in each toss of a coin; defective or not defective light bulb Generally called “success” and “failure” Probability of success is p, probability of failure is 1 – p Constant probability for each observation e.g., Probability of getting a tail is the same each time we toss the coin Observations are independent The outcome of one observation does not affect the outcome of the other

40 Examples of Binomial Distribution Settings A manufacturing plant labels items as either defective or acceptable A firm bidding for contracts will either get a contract or not A marketing research firm receives survey responses of “yes I will buy” or “no I will not” New job applicants either accept the offer or reject it

41 P(X) = probability of X successes in n trials, with probability of success p on each trial X = number of ‘successes’ in sample, (X = 0, 1, 2,..., n) n = sample size (number of trials or observations) p = probability of “success” P(X) n X! nX p(1-p) X n X ! ()!    Binomial Distribution Formulas Mean Variance and Standard Deviation

42 In class exercise #45: Redo ICE #44 using the binomial formulas.

43 Using Binomial Tables (like Table 6 in back of book) n = 10 x…p=.20p=.25p=.30p=.35p=.40p=.45p= ………………………………………………………… …p=.80p=.75p=.70p=.65p=.60p=.55p=.50x Examples: n = 10, p =.35, x = 3: P(x = 3) =.2522 n = 10, p =.75, x = 2: P(x = 2) =.0004

44 In class exercise #46: Check your probabilities for ICE #44 using Table 6 in the back of the book.

45 In class exercise #46: Check your probabilities for ICE #44 using Table 6 in the back of the book.

46 In class exercise #47: An important part of the customer service responsibilities of a telephone company relates to the speed with which customer service troubles can be repaired. Suppose past data indicate that the likelihood is 0.70 that troubles in residential service can be repaired on the same day. For the first five troubles reported on a given day, what is the probability that a) all five will be repaired on the same day? b) at least three will be repaired on the same day?

47 Table 6 again

48 In class exercise #48: If I shoot three point baskets with 68% accuracy, what is the probability I make exactly 7 out of 10?

49 Using Excel to get Binomial Probabilities Example: n = 10, p =.68, x = 7: then Excel tells us P(7) =.2644