Please turn off cell phones, pagers, etc. The lecture will begin shortly.

1. Please turn off cell phones, pagers, etc. The lecture will begin shortly. There will be a very easy quiz at the end of today’s lecture.

2. Announcements • Exam 3 will be held next Wednesday, April 5. • It will cover material from Lectures 20-27, • or chapters 12, 13 and 16. • Each student may bring one sheet of notes • (8.5 × 11 inches, both sides) to use during the • exam • Monday’s class (Lecture 28) will be a review and • cover example questions.

3. Lecture 26 This lecture will cover select topics from Chapter 16. • What is probability? (Section 16.2-16.3) • Rules of probability (Section 16.4)

4. 1. What is probability? Probability is a way to describe the likelihood of an uncertain event in advance, before we observe whether it happens or not. Statisticians use capital letters at the beginning of the alphabet (A, B, …) to symbolize events. If we denote a particular event by A, then the probability that A will occur is written as P(A) P(A) = probability that A will occur P(A) = 1 means that the event will definitely happen. P(A) = 0 means that the event will definitely not happen. But what is the meaning of probabilities between 0 and 1?

5. First interpretation: Relative Frequency The relative frequency interpretation of probability is: P(A) = proportion of the time that A occurs in the long run, if the “experiment” is repeated under identical conditions. (Here, “experiment” is used broadly to indicate any kind of action for which the result is not known in advance.) For example, suppose the experiment is to toss a coin, and suppose the event A is “head.” In this case, P(A) = 0.5 means that, if we were to toss the coin a very large number of times, the proportion of tosses for which we would observe “head” would be 0.5 or 50%.

6. tosses heads proportion 10 4 .400 100 56 .560 Example Using my computer, I simulated tosses of a fair coin. 1,000 464 .464 10,000 4,960 .496 100,000 49,740 .497 1,000,000 500,358 .500 As the number of tosses increases, the proportion of the time that “head” occurs gets closer and closer to 0.5. This supports the claim that P(A) = 0.5.

7. Another example Here the experiment is to roll a pair of dice, and the event A is to roll a 7. rolls sevens proportion 10 1 .100 100 12 .120 1,000 158 .158 10,000 1,660 .166 100,000 16,304 .163 1,000,000 165,882 .166 As the number of rolls increases, the proportion of the time that “seven” occurs is getting closer to 1/6 = 0.167. This supports the claim that P(A) = 0.167.

8. Comments on the Relative Frequency interpretation • This is the most widely accepted interpretation • of probability • It is regarded as objective, because answers can be • verified (e.g. by computer simulation) • This interpretation makes sense for experiments that • can actually be repeated under similar conditions • It does not make as much sense for special or • one-time situations that cannot be repeated

9. Second interpretation: Personal Probability The personal probability interpretation is: P(A) = degree to which an individual believes that A is going to happen This value will obviously differ from one person to another. Individual’s probabilities may differ because • they have varying amounts and kinds of knowledge • people are not equally good at assessing uncertainty

10. Example Recent report by the Environmental Protection Agency: “Global warming is most likely to raise sea level 15 cm by 2050 and 34 cm by 2100…There is a 1% chance that global warming will raise sea level 1 meter in the next 100 years.” Senator John Kerry (March 17, 2006): “I can say to absolute certainty that if things stay exactly as they are today…within the next thirty years, the Arctic ice sheet is gone…If that melts, you have a level of sea level increase that wipes out Boston harbor, New York harbor.”

11. Comments on the Personal Probability interpretation: • For evaluating the risks of rare or one-time events, • this may be the only way • Many subjective probabilities are simply an individual’s • statement of personal beliefs and biases • There are rigorous methods for combining • expert opinion and evidence to update subjective • probabilities • In retrospect, many evidence-based statements about • subjective probability have been very wrong • (e.g. O-ring failure in the 1986 Challenger explosion)

12. 2. Rules of Probability Probability follows certain rules. The purpose of these rules is that, if we know the probabilities of certain events, we will then be able to compute the probabilities of other events. Rule 0 The probability of any event must lie between 0 and 1 0 ≤ P(A) ≤ 1 (This rule is obvious. If you compute a probability and the answer is negative or greater than 1, you did something wrong.)

13. Rule 1: Complementary Events The probability that A does not happen is 1 minus the probability that it does happen. P(not A) = 1 – P(A) Example: If you roll a pair of dice, the probability of getting 7 or higher is 7/12 = .583. What is the probability of getting 6 or lower? “6 or lower” is the complement (i.e., the opposite) of “7 or higher.” P(6 or lower) = 1 – P(seven or higher) = 1 - .583 = .417

14. Mutually exclusive events Two events are said to be mutually exclusive if they cannot happen together. For example, the two events A: subject is 9 years old B: subject has been divorced are mutually exclusive. The two events A: subject died from a heart failure B: subject had hepatitis are not mutually exclusive.

15. Rule 2: Addition rule If events A and B are mutually exclusive, then P(A or B) = P(A) + P(B) Example: If you roll a single die, what is the probability of getting a 5 or 6? Are the events “roll a 5” and “roll a 6” mutually exclusive? Yes. P(5 or 6) = P(5) + P(6) = 1/6 + 1/6 = 1/3.

16. Independent events Two events are said to be independent if knowing whether or not one event occurred does not change the probability of the other event occurring. For example, suppose you buy one lottery ticket each week. The events A: win the lottery this week B: win the lottery next week are independent, because winning or losing this week does not change your chances of winning next week. The two events A: subject is female B: subject has breast cancer are not independent, because the risk of breast cancer is much higher for women than for men.

17. Rule 3: Multiplication rule If the events A and B are independent, then P(A and B) = P(A) × P(B) Example: If you roll a pair of dice, what is the probability that you get a 12? The only way to get a 12 is to get 6 on the first die and 6 on the second die. Are these events independent? Yes. P(6 on first and 6 on second) = P(6 on first) × P(6 on second) = 1/6 + 1/6 = 1/36.

18. More complicated example If there are 200 people in this room, what is the probability that today is someone’s birthday? (Assume each person has probability 1/365 of having a birthday today, independently of any other person). P(today is someone’s birthday) = 1 – P(no one was born today) = 1 – P(person 1 not born today and person 2 not born today … and person 200 not born today) = 1 – (364/365) × (364/365) … × (364/365) = .578