Probability Rules

Review of Basic Probability Rules Complement Rule P(A) = 1 – P(Ac) Addition Rule for Mutually Exclusive Events P(A or B) = P(A) + P(B) Multiplication Rule for Independent Events P(A and B) = P(A)*P(B) “At Least One” Rule P(At least one) = 1 – P(none)

Venn Diagram: Mutually Exclusive Events One card is drawn from a standard deck of cards. What is the probability that it is an ace or a nine? A B Ace nine Events A and B are mutually exclusive. A card can be either an Ace or a nine, but can not be both

Venn Diagram: Events that are Not Mutually Exclusive One card is drawn from a standard deck of cards. What is the probability that it is red or an ace? Red Ace Both red and an ace These events are not mutually exclusive as it is possible for a card to be both red and an ace (ace of hearts, ace of diamonds)

Probabilities of Events that Are Not Mutually Exclusive Find P(Red or Ace): 0.5 0.077 Red Ace 0.038 = P(Red) + P(Ace) – P(Both Red and Ace) = 0.5 + 0.077 -0.038 =0.539

General Addition Rule Used when events are not mutually exclusive. P(A or B) = P(A) + P(B) – P(A and B) Example: The Illinois Tourist Commission selected a sample of 200 tourists who visited Chicago during the past year. The survey revealed that 120 tourists went to the Sears Tower, 100 went to Wrigley Field and 60 visited both sites. What is the probability of selecting a person at random who visited both the Sears Tower and Wrigley Field?

General Addition Rule (continued) P(Sears Tower) = 120/200 = 0.6 P(Wrigley Field) = 100/200 = 0.5 P(Both) = 60/200 = 0.3 0.3 0.6 0.5 S W P(S or W) = P(S) + P(W) – P(S and W) = 0.6 + 0.5 - 0.3 = 0.8

General Addition Rule (continued) What is the probability that a randomly selected person visited either the Sears Tower or Wrigley Field but NOT both? P(S or W but NOT both) = P(S or W) – P(S and W) = 0.8 – 0.3 = 0.5 Second approach: P(S and Wc) = 0.6 – 0.3 = 0.3 P(W and SC) = 0.5 – 0.3 = 0.2 P(S or W but NOT both) = P(S and Wc) + P(W and SC) = 0.3 + 0.2 = 0.5

General Addition Rule (continued) What is the probability that a randomly selected tourist went to neither location? P(neither location) = 1 – P(either location) = 1 – P(S or W) = 1 – 0.8 = 0.2

Conditional Probabilities Here is a contingency table that gives the counts of ECO 138 students by their gender and political views. (Data are from Fall 2005 Class Survey) P(Female) = 77/137 = 0.562 P(Female and Liberal) = 30/137 = 0.219 What is the probability that a selected student has moderate political views given that we have selected a female?

Conditional Probabilities (continued) What is the probability that a selected student has moderate political views given that we have selected a female? P(Moderate | Female) = 24/77 = 0.311 Conditional probability, P (B|A) – the probability of event B given event A.

Conditional Probabilities (continued) Formal Definition: P(B|A) = P(A and B) P(A) Example: P(Moderate and Female) P(Female) =(24/137) / (77/137) = 0.175 / 0.562 = 0.311

Multiplication Rule Multiplication Rule for Independent events: P(A and B) = P(A) * P(B) Independent – the occurrence of one event has no effect on the probability of the occurrence of another event. Example: A survey by the American Automobile Association (AAA) revealed that 60 percent of its members made airline reservations last year. Two members are selected at random. What is the probability both made airline reservations last year? P(R1 and R2) = P(R1)*P(R2) = (0.6)*(0.6) = .36

General Multiplication Rule Use when events are Dependent. P(A and B) = P(A) * P(B|A) For two events A and B, the joint probability that both events will happen is found by multiplying the probability event A will happen by the conditional probability of event B occurring.

General Multiplication Rule (continued) Example: A county welfare agency employs 10 welfare workers who interview prospective food stamp recipients. Periodically the supervisor selects, at random, the forms completed by two workers to audit for illegal deductions. Unknown to the supervisor, three of the workers have regularly been giving illegal deductions to applicants. What is the probability that both of the two workers chosen have been giving illegal deductions?

General Multiplication Rule (continued) Solution: Define the following two events: A = First worker selected gives illegal deductions B = Second worker selected gives illegal deductions We want to find the probability that both A and B occur. To find the P(A) consider the following Venn Diagram. I = worker with illegal deductions N = worker not giving illegal D Each observation in the sample space is equally likely. P (A) = P(I1) + P(I2) + P(I3) = 1/10 + 1/10 + 1/10 = 3/10 or 0.30 N1 N2 N3 N4 N5 N6 N7 I1 I2 I3 A

General Multiplication Rule (continued) To find the conditional probability, P(B|A), we need to make changes to the sample space. Remember our assumption is that the first worker selected is giving illegal deductions. P (B|A) = P(I1) + P(I2) = 1/9 + 1/9 = 2/9 Substituting P(A) and P(B|A) into the formula for the general multiplication rule, we find P(A and B) = P(A)P(B|A) = (3/10) * (2/9) = 6/90 = 1/15 or 0.067 N1 N2 N3 N4 N5 N6 N7 I1 I2 B|A

Tree Diagram A tree diagram is a display of conditional events or probabilities that is helpful in thinking through conditioning. N (6/9) N and N = (7/10)(6/9) = 42/90 N (7/10) N and I = (7/10)(3/9) = 21/90 I = (3/9) (3/10) N =(7/9) I and N = (3/10)(7/9) = 21/90 I I = (2/9) I and I = (3/10)(2/9) = 6/90

Independent Events? Again, events are independent when the outcome of one event does not influence the probability of the other. Events A and B are independent whenever P(B|A) = P(B) In the case of independent events the general multiplication rule reduces to the simple multiplication rule. P(A and B) = P(A) * P(B|A) = P(A) * P(B)

Exploring Independence Is the probability of being liberal independent of gender for ECO 138 students? In other words, does P(Liberal | Female) = P(Liberal)? P(Liberal|Female) = 30/77 = 0.39 P(Liberal) = 47/137 = 0.343 Because these probabilities are not equal, we can be pretty sure that liberal political views are not independent of the student’s gender