Lecture 14

3/6/08Lecture 142 Birthday Problem In a classroom of 21 people, what is the probability that at least two people have the same birthday? Event A: at least two people have the same birthday out of the 21 people. A C : every person has a different birthday out of the 21 people. P(A)=1-P(A C ) =1-(365/365)(364/365)…(345/365)

3/6/08Lecture 143

Birthday problem What about the probability of exactly one pair? n*(n-1)/2* (365/365)(1/365)(364/365)…(365-n+2)/365

Monte Hall Problem 3 doors, one prize – Select one door – Host show opens one of the other two doors that do not contain the prize – You are given a chance to keep the door you selected or switch to the other non-open door. What shall I do?

Play on-line html html

Analysis Assumptions: – Initially, each door has the same chance to contain the price – If selected door contains the price, Monty selects the door to open at random with equal probability

Setup is important I can relabel the doors: – M – the one I selected – L – left door out of the remaining – R – right door out of the remaining P(Prize in M)=P(prize in L)=P(prize inR)=1/3 Two events: Open L, Open R – We need P(Prize in M | Open L)

Calculation Draw a tree – explain the situation

Modifications Possible modification: – Monty favors a door: What changes is P(Open L | Price in R) ≠ 1/2 – Monty can goof (open a door with the price in it) The tree changes In any case switching never hurts

Limitation of mean When evaluating games – we often looked at the mean gain as a proxy for understanding the game This might be insufficient – In magamillions and powerball the jackpot sometimes rises so high that the average gain is positive. Q: Is it rational to play? – Issues: Adjustment for ties (drops down expected gain significantly) How many games one needs to play before winning?

Let’s design a Lottery! How to make a lottery? – Define random generating mechanism – Define payoffs Makes money on average Risk is not too bad How much reserves are needed?

Formats of games Genoese type – Draw m balls out of M; players also select m numbers UK National lottery 6/49 NC Cash 5: 5/39 (most prices are pari-mutuel) Powerball 5/59&1/35 (most prices with fixed, jackpot pari- mutuel) Keno type – Draw m balls out of M with players select k numbers Number type – m digits (0,1,…,9) drawn with replacement – players try to match numbers in order or out of order NC pick 3, NC pick 4

Prices Fixed price – the winning is determined ahead of time – Simpler to understand / higher risk for lottery Pari-mutual – the winners split a predetermined portion of the pot – Harder to sell / no risk to lottery