Chapter 2

1. Chapter 2 Probability Concepts and Applications

2. Probability • A probability is a numerical description of the chance that an event will occur. • Examples: P(it rains tomorrow) P(flooding in St. Louis in September) P(winning a game at a slot machine) P(50 or more customers coming to the store in the next hour) P(A checkout process at a store is finished within 2 minutes)

3. Basic Laws of Probabilities • 0 <= P(event) <= 1 • Sum of the probabilities of all possible outcomes of an activity (a trial) equals to 1.

4. Subjective Probability • Subjective Probability is coming from person’s judgment or experience. • Example: • Probability of landing on “head” when tossing a coin. • Probability of winning a lottery. • Chance that the stock market goes down in coming year.

5. Objective Probability • Objective Probability is the frequency that is derived from the past records • How to calculate frequency? • Example: page 23 and page 34

6. Example, p.25 (a)Calculate probabilities of daily demand from data in the past

7. Example, p.25 (b)

8. ‘Possible Outcomes’ vs. ‘Occurrences’ • In the given data, differentiate the column for ‘possible outcomes’ of an event from the column for ‘occurrences’ (how many times an outcome occurred). • Probabilities are about possible ‘outcomes’, whose calculations are based on the column of ‘occurrences’.

9. Union of Events • Union of two events A and B refers to (A or B), which is also put as AUB. • For example, drawing one from 52 playing cards. If A= a “7” is drawn, B= a “heart” is drawn, then AUB means “the card drawn is either a ‘7’ or a ‘heart’”.

10. Intersection of Events • Intersection of two events A and B refers to (A and B), which is also put as A∩B or simply AB. • For example, drawing one from 52 playing cards. If A= a “7” is drawn, B= a “heart” is drawn, then A∩B means “the card drawn is ‘7’ and a ‘heart’”.

11. Conditional Probability • A conditional probability is the probability of an event A given that another event B has already happened. • It is put as P(A|B). • For example, • P(a man has got cancer | his PSA test value is 1.5), • P(battery is dead | engine won’t start)

12. Formulas for U and ∩ • P(AUB) = P(A) + P(B)  P(A∩B) • P(A∩B) = P(A|B) * P(B) by algebraic rule we have P(A|B) = [P(A∩B)] / P(B)

13. Example (p.27-28) • Randomly draw one from 52 playing cards. Let A= a “7” is drawn, B= a “heart” is drawn: • P(A) = 4/52, P(B) = 13/52, • P(A∩B) = P(AB) = 1/52. • P (AUB) = 4/52 + 13/52  1/52 = 16/52 • P(A|B) = [P(AB)] / P(B) = [1/52] / [12/52] = 1/13.

14. Mutually Exclusive Events • Events are mutually exclusive if only one of the events can occur on any trial. • If A and B are mutually exclusive, then P(A∩B) = 0.

15. Examples • Mutually exclusive: • (it rains at AC; it does not rain at AC) • Result of a game: (win, tie, lose) • Outcome of rolling a dice: (1, 2, 3, 4, 5, 6) • NOT mutually exclusive: • (a randomly drawn card is a ‘7’; a randomly drawn card is a ‘heart’.) • (one involves in an accident; one is hurt in an accident)

16. Probabilities for Mutually Exclusive Events • If events A and B are mutually exclusive, then: P(AUB) = P(A) + P(B)

17. Independent Events • Two events are independent if the occurrence of one event has no effect on the probability of occurrence of the other. • If A and B are independent, then P(A|B) = P(A), and P(B|A) = P(B).

18. Examples of Independent Events • (results of tossing a coin twice) • (lose $1 in a run on a slot machine, lose another $1 in the next run on the slot machine) • (it rains at AC; it does not rain at LA)

19. Examples for Non-Independent Events • (your education; your starting salary) • (it rains today; there are thunders today) • (heart disease; diabetes); • (losing control of a car; the driver is drunk).

20. Formulas for P(A∩B) if A and B Are Independent • If A and B are independent, then their joint probability formula is reduced to: P(A∩B) = P(A) * P(B)

21. Example • Drawing balls one at a time with replacement from a bucket with 2 blacks (B) and 3 greens (G). • Is each drawing independent of the others? • P(B) = • P(B|G) = • P(B|B) = • P(GG) = • P(GBB) =

22. Example • Drawing balls one at a time without replacement from a bucket with 2 blacks (B) and 3 greens (G). • Is each drawing independent of the others? • P(B) = • P(B|G) = • P(B|B) = • P(GB) = • P(G|B) =

23. Discerning between Mutually Exclusive and Independent • A and B are mutually exclusive if A and B cannot both occur. P(A∩B)=0. • A and B are independent if A’s occurrence has no influence on the chance of B’s occurrence, and vice versa. P(A|B)=P(A) and P(B|A)=P(B).

24. Discerning Conditional Probability and Joint Probability • Joint probability P(AB) or P(A∩B) is the chance both A and B occurs before either actually occurs. • Conditional probability P(A|B) is the chance of A after knowing that B has occurred.

25. Random Variable • A random variable is such a variable whose value is selected randomly from a set of possible values.

26. Examples of Random Variables • Z = outcome of tossing a coin (0 for tail, 1 for head) • X=number of refrigerators sold a day • X=number of tokens out for a token you put into a slot machine • Y=Net profit of a store in a month • Table 2.5 and 2.6, p.33

27. Probability Distribution • The probability distribution of a random variable shows the probability of each possible value to be taken by the variable. • Example: P.34, P.35, P.38.

28. Example, Table 2.7, p.34X = the quiz score of a student

29. Expected Value of a Random Variable X • The expected value of X = E(X): where Xi=the i-th possible value of X, P(Xi)=probability of Xi, n=number of possible values. • E(X) is the sum of X’s possible values weighted by their probabilities.

30. Interpretation of Expected Value • The expected value is the average value (mean) of a random variable.

31. Xi, P(Xi), and E(X) in Example p.34

32. Other Examples • Expected value of a game of tossing a coin. • Expected value of playing with a slot machine (see the handout).

33. Standard Deviation of X • Standard deviation (SD), , of random variable X is the average distance of X’s possible values X1, X2, X3, … from X’s expected value E(X).

34. Variance of X • To calculate standard deviation (SD), we need to first calculate “variance”. • Variance 2 = (SD)2. • SD =  =

35. Standard Deviation and Variance • Both standard deviation and variance are parameters showing the spread or dispersion of the distribution of a random variable. • The larger the SD and variance, the more dispersed the distribution.

37. Calculating Variance 2 • where • n=total number of possible values, • Xi=the i-th possible value of X, • P(Xi)=probability of the i-th possible value of X, • E(X)=expected value of X.

38. Calculating 2 in Example p.34

39. Normal Distribution • The normal distribution is the most popular and useful distribution. • A normal distribution has two key parameters, mean  and standard deviation . • A normal distribution has a bell-shaped curve that is symmetrical about the mean .

40. Standard Normal Distribution • The standard normal distribution has the parameters =0 and =1. • Symbol Z denotes the random variable with the standard normal distribution