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Chapter 2

Chapter 2. Probability Concepts and Applications. 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)

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Chapter 2

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

  5. Example, p.23 (A)Calculate probabilities of daily demand from data in the past

  6. Example, p.23 (B)

  7. ‘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’.

  8. Example, p.34Calculate probabilities of possible responses from the sampled data

  9. 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.

  10. Mutually Exclusive Events • Events are mutually exclusive if only one of the events can occur on any trial. • Examples: • (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) • The follows are NOT mutually exclusive: • (it rains at AC; it does not rain at LA) • (one involves in an accident; one is hurt in an accident)

  11. Probabilities for Mutually Exclusive Events • If events A and B are mutually exclusive, then: P(A and B) = 0 P(A or B) = P(A) + P(B)

  12. Independent Events • Two events are independent if the occurrence of one event has no effect on the probability of occurrence of the other. • Examples: • (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) • (results of tossing a coin twice)

  13. Examples for Non-Independent Events • Non-independent events are called dependent events. Examples: • (your education; your income level) • (it rains today; there are thunders today) • (heart disease; diabetes); • (losing control of a car; the driver is drunk).

  14. Probability Formulas for Independent Events • If event A and event B are independent, then: P(A given B) = P(A|B) = P(A); P(A and B) = P(AB) = P(A) * P(B)

  15. Example 4, page 26 • 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) =

  16. Formula for Dependent Events • P(AB) = P(A)*P(B|A) = P(B)*P(A|B) where P(B|A) means probability of B given A.

  17. 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) =

  18. Discerning between Mutually Exclusive and Independent • A and B are mutually exclusive if A and B cannot both occur. P(AB)=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).

  19. Random Variable • A random variable is such a variable whose value is selected randomly from a set of possible values. • The meaning of a random variable must be clearly defined.

  20. 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 2.4 and 2.5, p.33-34

  21. 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.

  22. Distribution for textbook question(p.34)

  23. 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.

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

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

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

  27. 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).

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

  29. 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.

  30. 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.

  31. Calculating 2 in Example p.34

  32. 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 .

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

  34. Calculate Probabilities on Normal Distribution • Suppose X follows the normal distribution with mean  and standard deviation . We want to find P(X<x), where x is a given number • Step 1. Calculate • Step 2. Find P(Z<z) by using Table 2.9 on page 45.

  35. Example, page 47 • We want to find P(X<130) where X follows Normal(100, 15), i.e. =100, =15. • To solve the problem: Step 1. Step 2. From Table 2.7, P(Z<2)=0.997. So, P(X<130)=0.997=97.7%.

  36. Example, Page 46-47 (1 of 2) • Suppose X follows Normal(100, 20), i.e. =100, =20. • We want P(X<75). • To solve it: Step 1. Step 2. P(X<75) = P(Z<1.25) =P(Z>1.25) =1P(Z<1.25)=10.89435=0.10565.

  37. Example, Page 46-47 (2 of 2) • To find P(110<X<125) • Solving the problem: * Note that P(110<X<125)=P(X<125)  P(X<110) * For P(X<125), and P(X<125)=P(Z<1.25)=0.89435 * For P(X<110), and P(X<110)=P(Z<0.5)=0.69146 * P(110<X<125)=0.894350.69146=0.20289

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