Multiplication Rule We now discuss the situation when both events A and B occur.

Example: Suppose that you toss a balanced coin twice. You are looking for the probability of getting two heads. A = first toss is a head B = second toss is a head Note we are looking at two different flips of the coin. A and B are not disjoint since we are looking at them both occurring. We are looking for the combination of two separate events and thus need to flip the coin twice. We are looking for P(A and B).

Draw a Venn diagram for this event.

Multiplication rule: P(A and B) = P(A) P(B) This is the multiplication rule for independent events. What is P(A getting a head) ? What is P(B getting a head) ? So what is the probability of P(A and B) ? Why do you think events A and B must be independent for the above rule to hold?

Draw a tree diagram of event (A and B) and use this to support your conclusion P(A and B) = 1/4

Disjoint events VS Independent events Let A = rolling a die and getting a 5 Let B = rolling the die and getting a 3. Let A or B = the event of rolling a 3 or 5 Let A and B = the event of rolling a three and then a 5 What is the probability of each of the above? From what you answered above are independent events considered disjoint? Why?

Can we always say that disjoint events are not independent? Why? Note: The multiplication rule we have given works only for independent events. We will look at the multiplication rule for events that are not independent in the next section.

Problem 28 An automobile manufacturer buys computer chips from a supplier. The supplier sends a shipment containing 5% defective chips. Each chip chosen from this shipment has probability.05 of being defective, and each automobile uses 12 chips selected independently. What is the probability that all chips in a car will work properly?

6.33 Telephone Success Most sample surveys use random digit dialing equipment to call residential telephone numbers at random. The telephone polling firm Zogby International Reports that the probability that a call reaches a live person is.2. calls are independent. a)A polling firm places 5 calls. What is the probability that none of them reaches a person? b) When calls are made to New York City, the probability of reaching a person is only.08. What is the probability that none of 5 calls made to New York City reaches a person? Feldts c) From part b above, what is the probability at least one call reaches a person?

Examples from book: Screening large numbers of blood samples for HIV, the virus that causes AIDS, uses an enzyme immunoassay (EIA test that detects antibodies to the virus. Samples that test positive are retested using a more accurate western blots test. Applied to people who have no HIV antibodies, EIA has probability about.006 of producing a false positive. If the 140 employees of a medical clinic are tested and all 140 are free of HIV antibodies, What is the probability that at least one false positive will occur? We will assume that as part of the probability model that the test results for different individuals are independent. What is the probability that a single person tests positive when they do not have HIV? This is a false positive. What are all of the possible outcomes for the group of 140 employees? What is the P(a negative result for a single person)? This is called the complement of a positive result. What is the complement of at least one person of the 140 people that do not have HIV to test positive)?