Probability Concepts and Applications 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

Quantity Demanded (Gallons) Example, p.25 (a) Calculate probabilities of daily demand from data in the past Quantity Demanded (Gallons) Number of Days 40 1 80 2 50 3 20 4 10 What is the probability that daily demand is 4 gallons? 3 gallons? 2 gallons? …

Quantity Demanded (Gallons) Number of Days Frequency as Probability Example, p.25 (b) Quantity Demanded (Gallons) Number of Days Frequency as Probability 40 1 80 2 50 3 20 4 10 Total

‘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: NOT 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): 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 Xi P(Xi) i X’s possible value Probability 1 5 0.1 2 4 0.2 3 0.3

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 =  =

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 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 P(Xi) i X’s possible value Probability 1 5 0.1 2 4 0.2 3 0.3

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