MATHEMATICS E-102, FALL 2005 SETS, COUNTING, AND PROBABILITY Outline #1 (Probability, Intuition, and Axioms) Last modified: September 19, 2005 Reference: EP(Elementary Probability, by Stirzaker), Chapter 0 and sections 1.1 through 1.4. 1. Probability by Symmetry (EP section 0.3) For centuries gamblers have been estimating probabilities, frequently correctly, on the basis of the notion that there is no reason to expect any outcome to be more likely than any other. Here are some questions that are relevant to the games that will be available on Casino Night. In each case, determine the probability of the specified event by identifying a set of outcomes that seem equally likely and counting how many of them correspond to the outcome of interest. In blackjack, a deck of cards is shuffled thoroughly, and the dealer deals himself a card face-down. What is the probability that it is a spade? An ace? A 10-point card (10, Jack, Queen, or King)? An American roulette wheel has 38 slots, numbered 1-36, 0, and 00. All appear to be equally likely resting places for the ball. What is the probability that the number in which the ball comes to rest is odd? In the range 1-6? Either 29 or 30? In craps, the shooter rolls two dice, which appear to be perfectly symmetrical. What is the probability that the shooter rolls 12(in which case your bet against her would be refunded to you?) Rolls 7? Rolls 11? Rolls either 7 or 11 and wins immediately? Rolls 2,3, or 12 ( craps ) and loses immediately? 2. Probability by Experiment(EP section 0.4) I will ask a randomly chosen member of the class whether he or she has previously taken a math course in the Harvard Extension School. Before I do, estimate the probability that the person will answer yes. I will ask 10 randomly chosen members of the class whether they have previously taken a math course in the Harvard Extension School. After hearing the results, estimate the probability that the eleventh person asked this question will answer yes. Would you be more confident in your estimate if there were 100 people for me to ask? Why? 1
Suppose that the question was instead Do you have a birthday during the months February-May inclusive? 3. Payoffs and Probability(EP section 0.5) In craps, the shooter rolls a 4, thereby establishing a point of 4. On the next roll you are invited to bet x dollars. If the shooter rolls 4, you win 12 dollars, If he rolls 7, you get nothing. Otherwise you get your x dollars back. What is a fair amount x to pay for the opportunity? Sam Adams and Peter Stuyvesant decide to make a friendly bet on the turn of a card from a well-shuffled deck. Sam puts up $3; Peter puts up $10. If the card is a face card (Jack, Queen, King), Sam gets the $13; otherwise Peter gets it. Why is this a fair bet? 4. Fair Price and Probability(EP section 0.6) Sam Adams and Peter Stuyvesant decide to make a friendly bet on the AL East pennant race. Sam puts up $15; Peter puts up $10. If the Red Sox win, Sam gets the $25; if the Yankees win, Peter gets it. They agree that this is a fair bet. What is their implied estimate of the probability that the Red Sox will win the division? A friend tells you that he thinks there is a 60% chance that the Red Sox will win the AL East and also expresses the opinion that there is one chance in four that Iraqis will reject their draft constitution. You ask, What do you think is the chance that both these things will happen? What is the reasonable answer? A friend tells you that she thinks there is a 30% chance that corporate profits will double in the next year and also expresses the opinion that there is one chance in three that the Dow will rise above 11000. You ask, What do you think is the chance that both these things will happen? What is a reasonable answer? 2
5. Notation to Combine Sets (EP, chapter 0 appendix) Define subsets of the U.S. population as follows: A actresses B actors C singers D basses E women F waitstaff Using the symbols for union (e.g. A B), intersection (e.g. A B), complement (e.g. A c ) and difference (e.g. A\B), express the following sets: thespians waitresses men tenors 6. Venn diagrams (EP, chapter 0 appendix) Give a Venn diagram proof that (A B) (A C) = A (B C) Prove the above identity without using a Venn diagram, by showing that the set on each side of the equals sign is a subset of the set on the other side. 7. Events and Sample Spaces(EP section 1.1). Give a couple of real-world examples of experiments, each with an associated sample space Ω. For each, specify an individual outcome in Ω to which you might assign a probability and an event (subset of Ω) that is not just an individual outcome. On what basis might you assign these probabilities? 3
8. Event Spaces(EP section 1.2). Derive formulas for the intersection and difference of sets A and B in terms only of union and complement. Illustrate the formulas using a Venn diagram. (EP, page 31) State the definition of an event space and explain why the closure requirements in the definition are sufficient to prove closure under difference and intersection. Give a couple of trivial examples of event spaces, and exhibit a collection of subsets of {1,2,3,4} that is not an event space. Suppose that the sample space Ω is the set of fictional murders and it includes as events the victim was a man and the method was poison. What other events must be included in order to have an event space? Give examples of other events that could be included in the event space but do not have to be. 9. Addition of Probabilities(EP section 1.3). State the basic addition rule for probability (2 disjoint events) and extend it by induction to n disjoint events. State the extension to a (countably) infinite set of disjoint events. A fortune teller says The probability that you will meet and marry your future spouse this year is 1. If it does not happen this year, 4 the probability that it will happen next year is 1, the year after 8 that 1, etc. While you remain single, you are immortal. 16 Use the extended addition rule to calculate the probability that you eventually marry. 4
10. Probability Functions (EP section 1.3). Define probability distribution and probability function ). Define probability space, and illustrate the definition by describing the probability space associated with rolling a single die. Prove that for a probability function P, P(B \ A) = P(B) P(A) if A B. Prove that for a probability function P, P(A) P(B) if A B. 11. Inclusion-Exclusion Rule(EP section 1.4). Prove that for a probability function P, P(A B) = P(A) + P(B \ A) = P(A) + P(B) P(A B) and illustrate this theorem with a diagram (EP figure 1.2) where Ω is the set of points in a square region of the blackboard. 12. Many Ways to Skin a Cat Calculate the probability, when a pair of cards are dealt from a wellshuffled single deck of cards, that at least one of them is a spade. Let A be the event that the first card dealt is a spade (and the second can be anything). Let B be the event that the second card dealt is a spade (and the first can be anything). Show that you get the same answer, 15, 34 by each of the following approaches. State in words, and illustrate with a Venn diagram, the reasoning behind each of the four approaches. (a) P(A B) = 1 P(A c B c ) (b) P(A B) = P(A) + P(B) P(A B) (c) P(A B) = P(A) + P(A c B) (d) P(A B) = P(A B c ) + P(A c B) + P(A B) 5
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