1. QR 32 Section #6 November 03, 2008 TA: Victoria Liublinska vliublin@fas.harvard.edu

2. Homework & Midterm comments Midterm comments: Solution is posted online Distribution of scores: mean 63.4 SD 7.3 median 65 Initial Project Proposals!

3. Outline for today: What is probability? Terminology and rules Disjoint vs. independent events Example of calculating probabilities and expectation for a random variable

4. Interpretations of Probability

5. Probability: Long run frequency We can think of it as a long run average of the number of times an event occurs in e sequence of independent trials It is applicable only when the situation might be repeated numerous times (at least conceptually) The outcomes have same chances every time (no time dependence) We still can’t predict individual outcome, only the long- run average

6. Personal Probability (Decision making)

7. Practice Problem Suppose a particular outcome from a random event has a probability of 0.02. Which of the following statements represent correct interpretations of this probability? Provide justification. i) The outcome will never happen. ii) The outcome will happen two times out of every 100 trials, for certain. iii) The outcome will happen two times out of every 100 trials, on average. iv) The outcome could happen, or it couldn't, the chances of either results are the same.

8. Probability Terminology Terminology random phenomenon – an event whose individual outcomes are uncertain but there is a regular distribution in a large number of repetitions. Examples: Coin tossing and dice rolling The lottery and other games of chance Drawing a random sample from some population outcome: the value of one replication of a random experiment or phenomenon, Coin Tossing: H with one toss of a coin HTT with three tosses

9. Probability Terminology (cont.) sample space (labeled S): is the set of all possible outcomes of a random phenomenon Examples: 1. Toss a coin three times: S = {HHH,THH,HTH,…,TTT} 2. Face showing when rolling a six-sided die: S = {1,2,3,4,5,6} 3. Pick a real number between 1 and 20: S ={[1,20]} Event (labeled A): a set of outcomes of a random phenomenon. Examples: 1. The event A that exactly two heads are obtained when a coin is tossed three times: A={HHT,HTH,THH} 2. The number chosen from the set of all real numbers between 1 and 20 is at most 8.23: A = {[1,8.23]} 3. The result of the toss of a fair die is an even number: A = {2,4,6}

10. Events in Sample Spaces (more Terminology) The union of two events A and B is the event that either A occurs or B occurs or both occur: The intersection of two events A and B is the event that both A and B occur. The complement of an event A, A c, is the event that A does not occur and thus consists of outcomes that are not in A

11. Probability rules

12. Example – mandatory drug tests A mandatory drug test has a false-positive rate of 1.2% (or 0.012) Given 150 employees who are drug free, what is the probability that at least one will (falsely) test positive? P(At least 1 positive) = P(1 or 2 or 3... or 150 “+”) = 1 – P(None “+”) = 1 – P(150 “-”) P(150 negative) = P(1 “-”) 150 = (0.988) 150 = 0.16 P(At least 1 positive) = 1 – P(150 “-”) = 0.84

13. Disjoint vs Independent Two events are independent if the occurrence of one of the events gives us no information about whether or not the other event will occur; that is, the events have no influence on each other (no INTERACTION between events). (me/you being in Boston) Two events are disjoint (or mutually exclusive) if it is impossible for them to occur together (not COMPATIBLE) (me being in Boston or Las Vegas) Connection: If two events are mutually exclusive, they cannot be independent and vice versa ( if two events are independent then they cannot be mutually exclusive (disjoint)). (If I am in Boston – I know that I am not in Las Vegas)

14. Example There is a bag with 4 balls in it: 2 are red, and 2 are black You draw two balls out of the bag, one at a time. Define the events: A: the first ball drawn is black B: the second ball drawn is black What is the probability of A ? Are A and B independent?

15. Independent events Rule 5: Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent P(A and B) = P(A) P(B) Sometimes called the multiplication rule for independent events. Does knowing the results of flipping a fair coin once affect the chances of heads on a 2 nd flip?

16. Mean of a random variable If X is a discrete random variable with k possible values, its probability distribution is: Value x 1 x 2 x 3...x k Probability p 1 p 2 p 3...P k Mean (or “expected value”) of X, denoted μ X, is given by E(X) =  X = x 1 p 1 + x 2 p 2 +... + x k p k If X represents a measure on some member of a population, the E(X) is the population mean

17. Example A lottery has 100 tickets. Each ticket costs $10. Only one ticket is the “lucky” one and the cash prize for it is $500. Is it worthwhile for Mary to buy a lottery ticket? Answer the question by computing Mary’s expected gain

18. Practicing with events I have 5 M&M’s left in the bag: 2 blue, 2 red, and 1 yellow. I plan on eating two more, and leaving the others in the bag. 1.What is the sample space of outcomes for pairs of M&M’s that I will eat? 2.Describe the set of events for: A: eating two mismatched M&M’s (different colors). B: eating at least one blue M&M (they are my favorite). 3.What is the probability of A? B? 4.Are A & B disjoint? Why? 5.Describe (in terms of outcomes), D, the intersection of A & B? and the union, E? 6.What is the probability of D? of E? 7.Are A & B Independent? Why? 1212 1

19. Excel Demonstration