1. 1 Chapter 4, Part 1 Repeated Observations Independent Events The Multiplication Rule Conditional Probability

2. 2 Repeated Observations / Trials In this section, we discuss probabilities that arise when we observe: –The same random phenomenon more than once. –Two or more different random phenomena, each of which could effect the others. We will use Classical Probability—all outcomes are equally likely.

3. 3 Repeated Dice Rolling Q: I roll my 20-sided die. What is the probability of rolling a 20? –Ans: 1/20 = 0.05 = 5% Q: I roll my die a second time. What is the probability of rolling a 20? –Ans: The same as before: 1/20 = 0.05 = 5% Q: I roll my die twice. What is the probability that both rolls are 20?

4. 4 Repeated Dice Rolling DIFFICULT(?) QUESTIONS: –I roll my die twice. You see that the first roll is 20. What is the probability that the second roll is 20? –I roll my die twice. You see that the second roll is 20. What is the probability that the first roll was 20?

5. 5 Repeated Dice Rolling Note the difference between: –Probability that both rolls were 20 (400 possible outcomes). –Probability that one roll was 20, given the result of the other roll (20 outcomes). We usually assume that the two rolls “have no effect on one another,” but we don’t actually know this for certain.

6. 6 Selection Without Replacement Assume that my hat has 5 pink slips and 5 blue slips. I pick one slip, and then pick a second slip without replacing the first. Q: What is the probability that the first slip is blue? Q: What is the probability that the second slip is blue?

7. 7 Selection Without Replacement The second probability depends on the result of the first selection. –First slip is Blue: There are 4 Blue, 5 Pink slips left. The probability that the second slip is Blue is 4/9. –First slip is Pink: There are 5 Blue, 4 Pink slips left. The probability that the second slip is Blue is 5/9. This example illustrates what we will call a conditional probability.

8. 8 The probability of one event occurring, assuming that another event has occurred is called a conditional probability. Idea: Knowing/assuming that one event occurred might change the probability of a second event. This commonly happens when we “select without replacement.” The first result has an effect on the second (and later) results. Note that this didn’t happen with the multiple die rolls. The first result does not have an effect on the second (or so we assume). Conditional Probability

9. We use the notation P(B|A), read as “the conditional probability of B, given A.” –Note that the order of events matters!!! This means: We assume that event A has occurred, and use this knowledge to compute the (classical) probability of B. P(B|A) and P(A|B) will usually be different numbers. Order of events matters!!!

10. Conditional Probability Example I have 5 blue and 5 pink slips in the hat. I choose two slips, without replacing the first. P(2 nd is blue | 1 st is Pink) = ?? –Assuming the 1 st is pink (note the order of events), the hat contains 5 blue, 4 pink. –The conditional probability is 5/9. P(2 nd is blue | 1 st is Blue) = ?? –After the 1 st selection, the hat contains 4 blue, 5 pink. –The conditional probability is 4/9.

11. The Multiplication Rule Notation (only in section 4-4): –We observe a random phenomenon twice. –“A and B” means that event A occurs on the first observation, and event B occurs on the second observation. –While this is technically a correct use of “A and B,” it requires that each “outcome” consist of two separate observations. This will probably confuse you, so… –I will use the notation “A then B” to emphasize that we are considering two observations.

12. The Multiplication Rule For any two events A, B. P(A then B) = P(A) x P(B|A) Example: I choose two slips from the hat (without replacing the first). The probability that both slips are blue is: P(Blue then Blue) = (5/10)*(4/9) = 2/9.

13. The Multiplication Rule P(A then B) = P(A) x P(B|A) The probability that the 1 st is blue and the 2 nd is pink? P(Blue then Pink) = (5/10)*(5/9) = 5/18.

14. 14 The Actual Definition of Conditional Probability The conditional probability of B given A (in that order) is defined to be: This is the standard use of “A and B” discussed last time. “A then B” is just a special case of this.

15. 15 Classical Conditional Probability For conditional probability, we assume that one event (A) occurred, and use this information to compute the probability of a second event (B). If we make a single observation, and all outcomes are equally likely, there is a very simple method for computing conditional probability…

16. 16 Classical Conditional Probability For the result of a single observation, P(B|A) is given by: # of outcomes in “A and B” # of outcomes in A In other words, we take the number of outcomes satisfying BOTH conditions, and divide by the number of outcomes satisfying the GIVEN condition.

17. Examples I select a student from the class. Assume all outcomes are equally likely, compute: –P(Female | Row 1) –P(Male | Back Row) –P(Back Row | Male) –P(Back Row)

18. Dependent Events More often than not, knowing that one event occurred will change our expectations about a second event. In terms of conditional probability, this means that P(B) ≠ P(B|A). If this is the case, we say that the two events are dependent.

19. Independent Events But in some cases, knowing that one event occurred DOES NOT change our expectations about a second event. In terms of conditional probability, this means that P(B) = P(B|A). If this is true, we say the events are independent. An example is the 20-sided die: P(2 nd is 20) = P(2 nd is 20 | 1 st is 20).

20. Independent Events, Example I have 5 blue, 5 pink slips in the hat. I choose two slips, REPLACING THE FIRST before drawing the second. Compute the following: –P(2 nd is Blue) –P(2 nd is Blue | 1 st is Blue) You should find these to be the same number, so these are independent events.

21. The Multiplication Rule for Independent Events Original multiplication rule: P(A then B) = P(A) x P(B|A) If the events are independent, then P(B|A) is equal to P(B). The multiplication rule reduces to: P(A then B) = P(A) x P(B)