Is it possible to beat the lottery system?
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1 Is it possible to beat the lottery system? Michael Lydeamore The University of Adelaide Postgraduate Seminar, 2014
2 The story One day, while sitting at home (working hard)...
3 The story Michael Lydeamore Is it possible to beat the lottery system? Postgraduate Seminar, 2014
4 The story
5 The story The article had a simple problem: Playing in the lottery is expensive, chances of winning are small Tickets with high chances are expensive So here is a better scheme!
6 Section 1 Counting and Probability Background
7 Counting and Probability Background We denote the number of ways to choose k objects from n as ( ) n n! = k k!(n k)!
8 Counting and Probability Background Odds: 1 : 800 implies that you will win once, and not win 800 times.
9 Counting and Probability Background Odds: 1 : 800 implies that you will win once, and not win 800 times. Probability: P = imples that you will win once every 800 times.
10 Counting and Probability Background Odds: 1 : 800 implies that you will win once, and not win 800 times. Probability: P = imples that you will win once every 800 times. Prob. = 1 odds odds
11 Section 2 Lottery basics
12 Lottery basics Each person has a game (or some number of games) which have 6 numbers on them. Every week, lottery is drawn. 6 numbers are chosen randomly, followed by 2 supplementaries. In order to win, you need a certain combination...
13 Lottery basics Division Main numbers Supps
14 And here s why you never win... To win division 1, you need all 6 numbers correct. There are ( 45 6 ) different tickets. Only one of them is a winner. Thus, your odds of winning are 1 in 8, 145, 060.
15 And here s why you never win... To win division 1, you need all 6 numbers correct. There are ( 45 6 ) different tickets. Only one of them is a winner. Thus, your odds of winning are 1 in 8, 145, 060. Or the probability of winning is
16 And here s why you never win... To win division 2, you need 5 numbers and 1 supplementary. There are still ( ) 45 6 different tickets but now the probability of having a winner is ( )( ) 6 2 = 12, 5 1 So, your odds of winning are 1 in 678, 755.
17 And here s why you never win... To win division 2, you need 5 numbers and 1 supplementary. There are still ( ) 45 6 different tickets but now the probability of having a winner is ( )( ) 6 2 = 12, 5 1 So, your odds of winning are 1 in 678, 755. Or the probability of winning is
18 And it continues... Division Odds Prob. 1 8, 145, , ,
19 And it continues... Division Odds Prob. 1 8, 145, , , Roughly 1 of my Facebook friends will win division 5 if we all bought a ticket, while 1 of Kelli s Facebook friends would win division 4 if they all bought a ticket.
20 The winnings Higher divisions give a higher payout: Division Payout 1 1, 013, , , (Taken from 08/03/2014 X Lotto Draw)
21 Different ticket styles In order to improve your chances of winning (or their profits) you can buy different types of tickets. System 12: Choose 12 numbers, receive all combinations of those 12 numbers.
22 Different ticket styles In order to improve your chances of winning (or their profits) you can buy different types of tickets. System 12: Choose 12 numbers, receive all combinations of those 12 numbers. How many games? ( ) 12 = 924 games 6
23 Which means more probability... Division Odds Prob. 1 8,
24 You can also win multiple times! For example, should you get all 6 primary numbers and 2 supps, you win Division Amount
25 You can also win multiple times! For example, should you get all 6 primary numbers and 2 supps, you win Division Amount But, a System 12 ticket costs $
26 Section 3 The Problem Itself
27 The Problem Itself The article suggests instead of spending all your hard-earned money on a System 12, you should use an alternative scheme.
28 The Problem Itself Consider winning division 1. In a System 12 our probability is For the alternative, our odds are ( ) in = 1 : and so the probability is So we lose a lot of the coverage, but the reduced System 12 costs only $27.10.
29 So how does it work? To test which scheme is better, we could look at the probability and calculated expected winnings. E[W 12 ] = P(Div 1)Payout P(Div 6)Payout 6 cost = = E[W a ] = = So, the alternative scheme is better... but not great.
30 The thing is... There are many examples of people buying a ticket for the first time and immediately winning the big prize. The system is inherently stochastic Is it necessarily fair to look at the average case situation?
31 The thing is... There are many examples of people buying a ticket for the first time and immediately winning the big prize. The system is inherently stochastic Is it necessarily fair to look at the average case situation? Let s instead use some simulation to make sure.
32 Simulating Relatively simple algorithm: 1 Choose a set of 12 numbers, set up the 42 games in the alternative scheme. 2 For each lottery game: 1 Pick 6 numbers and 2 supplementaries, 2 Check how many fall in the system 12, 3 Check how many fall in each alternative game. 4 Update winnings vector, including cost. 5 Go to 2.
33 And this is the result x 107 System 12 Reduced System Winnings Numaber of games played x 10 5
34 And this is the result... 2 x System 12 Reduced System 2 Winnings Number of games played x 10 5
35 Possible Extensions / Future Work Need to think more about why the prediction doesn t match the simulation. But, simulation has shown us that the best scheme out of these two is...
36 Possible Extensions / Future Work Need to think more about why the prediction doesn t match the simulation. But, simulation has shown us that the best scheme out of these two is... Neither!
37 Possible Extensions / Future Work Need to think more about why the prediction doesn t match the simulation. But, simulation has shown us that the best scheme out of these two is... Neither! The 0 scheme (where we don t bet) is better!
38 Conclusion We have seen: The basics of counting, The basics of a lottery, Two alternative schemes for betting on lotteries, Validating results via a simulation, Importantly, we saw that the better way to win on the lottery is to, in fact, not bet at all.
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INFORMATION FOR OBSERVERS. Gaming Transactions (Agenda Paper 11(i))
30 Cannon Street, London EC4M 6XH, United Kingdom Tel: +44 (0)20 7246 6410 Fax: +44 (0)20 7246 6411 Email: iasb@iasb.org Website: www.iasb.org International Accounting Standards Board This observer note