1. Lecture 14

2. Project 2 Select and analyze a game of chance DUE: on November 10.
Use calculation, R or other mean to make QUANTITATIVE observations about the game
Write a 5 page paper. Include:
Description of the game
Your analysis
Conclusions
References
DUE: on November 10.
Come to my office to discuss what makes a suitable project!

3. Office hour Office hour this Wednesday only 10:30—11:30am
Sorry for any inconvenience.

4. Monty Hall Problem 3 doors, one prize What shall I do? 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?

5. Play on-line

6. 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

7. 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)

8. Calculation
Draw a tree – explain the situation

9. Modifications Possible modification: In any case switching never hurts
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

10. Lottery What do the people who design lotteries need to consider?
Random structure
Prices
Risk
Making it attractive for consumers

11. Formats of games Genoese type Keno type Number 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

12. 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

13. Risk With large fixed prizes – we have higher risk
Larger variance means that we need to keep money on hand to cover unusual occurrences
Will use computer to investigate

14. NC cash 5 - 5/39 Match Prize % of Prize Pool Odds 1 in
5 of 5 Pari-mutuel % 575,757.0
4 of 5 Pari-mutuel % 3,387.0
3 of 5 Pari-mutuel 9.74%
2 of 5 $ % 9.6
It costs $1 to play
The distribution of prices is skewed towards higher prizes. Because prices are pari-mutuel there is almost no risk to the lottery.

15. How many ways to select balls?
How many ways one can select m balls out of M?
When ordered:
M(M-1)(M-2)…(M-m+1)
Drop the order
M(M-1)(M-2)…(M-m+1)/{m(m-1)…2 1}

16. NC Pick 4
Select a 4 digit number (0000,…,9999). Win if matches number drawn
Cost $0.5 or $1 – prize $2500 or $5000
Expected payout $.5 for every $1 placed
Some other ways to place bets with the the same expected payout

17. Weird stuff Some people are winning way too often! Most likely issue:
Cashing tickets for someone else (usually to avoid debt payment)
Lottery operators have been caught lying to people when checking if a ticket is a winner

18. Our game Genoese Type How many people will play? Select the payouts
Select m/M
How many people will play?
Select the payouts
What is the proportion we target to pay out in prices (usually 50-60%)?
How much to roll over for jackpot?
Small prices – more predictable; people win more often
Large prices – less predictable; people get more excited about them

19. Making it attractive People prefer huge sums of money
Rollover in jackpot increases excitement
Rollover requires to have the jackpot split by winners (pari-mutual: a fixed amount is split)
Smaller prices are not usually split – each winner gets the same amount (this creates extra risk for the lottery)
The top price must be won often enough
This depends on number of players (target audience)
Usually there is some “good cause” for which the proceeds are targeted