Lecture 14

3/6/08 Lecture 14 2

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.

• AC: every person has a different birthday out of the 21 people.

• P(A)=1-P(AC) =1-(365/365)(364/365)…(345/365)

3/6/08 Lecture 14 3

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

• http://math.ucsd.edu/~crypto/Monty/monty.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