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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) 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) Basic Laws of Probabilities • 0 <= P(event) <= 1 • Sum of the probabilities of all possible outcomes of an activity (a trial) equals to 1. 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. Objective Probability • Objective Probability is the frequency that is derived from the past records • How to calculate frequency? – Example: page 23 and page 34 Example, p.25 (a) Calculate probabilities of daily demand from data in the past Quantity Demanded (Gallons) 0 1 2 3 4 Number of Days 40 80 50 20 10 What is the probability that daily demand is 4 gallons? 3 gallons? 2 gallons? … Example, p.25 (b) Quantity Demanded (Gallons) 0 1 2 3 4 Total Number of Days 40 80 50 20 10 Frequency as Probability ‘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’. 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’”. 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’”. 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) Formulas for U and ∩ • P(AUB) = P(A) + P(B) P(A∩B) • P(A∩B) = P(A) * P(B|A) by algebraic rule we have P(B|A) = [P(A∩B)] / P(A) 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. 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. 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) Probabilities for Mutually Exclusive Events • If events A and B are mutually exclusive, then: P(AUB) = P(A) + P(B) 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). 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) 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). 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) 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) = 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) = 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). 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. Random Variable • A random variable is such a variable whose value is selected randomly from a set of possible values. 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 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. Expected Value of a Random Variable X • The expected value of X = E(X): n E ( X ) X i P( X i ) i 1 X 1 P( X 1 ) X 2 P( X 2 ) ... X n P( X n ) 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. Interpretation of Expected Value • The expected value is the average value (mean) of a random variable. Xi, P(Xi), and E(X) in Example p.34 X=a student’s quiz score i 1 2 3 4 5 Xi X’s possible value 5 4 3 2 1 n E ( X ) X i P( X i ) i 1 P(Xi) Probability 0.1 0.2 0.3 0.3 0.1 Other Examples • Expected value of a game of tossing a coin. • Expected value of playing with a slot machine (see the handout). 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). Variance of X • To calculate standard deviation (SD), we need to first calculate “variance”. • Variance 2 = (SD)2. • SD = = variance 2 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. Calculating Variance 2 n [ X i E ( X )] P( X i ) 2 2 i 1 • 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. Calculating 2 in Example p.34 X=a student’s quiz score Xi X’s possible value 5 4 3 2 1 i 1 2 3 4 5 E ( X ) 2.9 n as calculatedon p.35. 2 [ X i E ( X )]2 P( X i ) i 1 P(Xi) Probability 0.1 0.2 0.3 0.3 0.1 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 . Standard Normal Distribution • The standard normal distribution has the parameters =0 and =1. • Symbol Z denotes the random variable with the standard normal distribution