A First Dive into Singapore Pools
Lotteries have long been called a “tax on the stupid”. While I sympathize with this simplistic characterization, I do think it left some interesting questions unasked:
- Which lottery game is better among others?
- Is the lottery truly unbeatable?
- Is there a way to improve expected values of winning?
To answer these questions, we must first quantify the “stupidity” of lotteries. In this post, we aim to do just that within the context of Singapore’s lottery games, utilizing principles of probability analysis. Rather than accepting the invincibility of the lottery system as a given, we roll up our sleeves and take a deeper look. Sometimes, as the saying goes, we may just find a real $100 note lying on the ground waiting to be picked up. Our motivation for this exercise is partly drawn from the story of a group of punters who managed to beat the Massachusetts state lottery in 2005, as recounted in Jordan Ellenberg’s book.
All information followed is up-to-date as of 2023-04-26. The information is not intended to be and do not constitute financial advice, investment advice, trading advice or any other advice.
Lay of the Land
Singapore Pools, the sole perator legally allowed to run lotteries in the city state, offers three games currenly. Ranked by popularity, they are:
We will tackle 4D and Singapore Sweep first. The reason is that they both have fixed prizes, which makes the analysis straightforward. TOTO, with its variable jackpot and more complex mechanism, warrants closer scrutiny separately.
Singapore Sweep
The rules can be found here.
Overall the goal is to pick a number from 1000000 to 4499999 (both inclusive) that matches drawn numbers in the same range. Each pick costs S$3. There are 133 draws with various prizes, ranging from S$2.3M to S$1,000. If a number wins multiple prizes, it is entitled to all of them. The same combination probably isn’t sold twice (see “Counter-party Risk” section below). Additionally, a separate mini prize is awarded if the last two digits match with one of the nine sets of drawn two digit number.
Given the expected value of the sum of several random variables is equal to the sum of their expectations, we can calculate each prize’s individual expected value (EV) and add them up for the total expected value. Here is a table summarizing the results:
| Prize | Prob | Number of Price | EV |
|---|---|---|---|
| $2,300,000 | 0.00003% | 1 | $0.66 |
| $500,000 | 0.00003% | 1 | $0.14 |
| $250,000 | 0.00003% | 1 | $0.07 |
| $10,000 | 0.00003% | 10 | $0.03 |
| $5,000 | 0.00003% | 10 | $0.01 |
| $3,000 | 0.00003% | 30 | $0.03 |
| $2,000 | 0.00003% | 30 | $0.02 |
| $1,000 | 0.00003% | 50 | $0.01 |
| $6 | 9% | $0.54 | |
| Total | $1.51 |
From this table, it’s clear that playing the Singapore Sweep is not a wise financial decision. With an expected loss of 49.6% for every S$3 spent, players are not likely to come out ahead.
4D
The rules can be found here.
Overall the goal is to pick a number from 0000 to 9999 (both inclusive) that matches drawn numbers in the same range. Each pick costs S$1. There are 23 draws with various prizes, ranging from S$3,000 to S$60. If a number wins multiple prizes, it is entitled to all of them, and multiple winners do not split prizes.
An interesting mechanism for 4D is that better can choose from two types of play: big bet or small bet. Big bet has more winning combinations but less prize money, ranging from S$2,000 to S$60; the opposite is true for small bet, with prize money ranging from S$3,000 to S$800. This means that when calculating EVs, two tables need to be drawn, as followed:
Big bet
| Prize | Prob | Number of Winning Combination | EV |
|---|---|---|---|
| $2,000 | 0.01% | 1 | $0.20 |
| $1,000 | 0.01% | 1 | $0.10 |
| $490 | 0.01% | 1 | $0.05 |
| $250 | 0.01% | 10 | $0.25 |
| $60 | 0.01% | 10 | $0.06 |
| Total | $0.66 |
Small bet
| Prize | Prob | Number of Winning Combination | EV |
|---|---|---|---|
| $3,000 | 0.01% | 1 | $0.30 |
| $2,000 | 0.01% | 1 | $0.20 |
| $800 | 0.01% | 1 | $0.08 |
| Total | $0.58 |
Again, from these tables, it is clear that playing 4D, big or small, is not a wise financial decision. The less bad of the two is big bet, with an expected loss of 34%. Assuming the player’s utility function does not prefer big payouts, the small bet is strictly worse than the big bet.
Counter-party Risk
Counter-party risk is the likelihood or probability that one of those involved in a transaction might default on its contractual obligation.
For Singapore Sweep and 4D, the prizes are not split in case of multiple winners. Without mitigation, this means that the lottery operator can face uncapped loss if there are multiple winning tickets. Is there a tail risk that the operator cannot pay out the prize?
Not for 4D, as the operator mitigated the risk by reserving the right to declare a combination “sold out” for 4D:
We may be unable to accept your bet if the number has reached its sales limit for that draw.
As for Singapore Sweep, punter cannot freely choose her combination:
The new selection formats allow players to choose either the first or last four digits of their set of numbers, but not both. Every ticket has a seven-digit number.
Singapore Pools is also introducing another new selection format called the “100 Sweep” that will allow punters to buy 100 sets of numbers in running sequence, with the last two digits ranging from 00 to 99. This will be printed on just one ticket slip.
The second option, 100 Sweep, is no longer offered as of April 23, 2023.
By giving only limited flexibility when choosing the combination, the operator prevents the same combination from being oversold, effectively curtailing its tail risk.
Conclusion
In this article, we delved into two of the three lottery games legally offered in Singapore and found that they are both losing propositions mathematically. To answer the questions we set out to investigate:
- Which lottery game is better among others?
- 4D big bet is the least bad among the three (Sweep, 4D big, 4D small). One is only expected to lose 34c per dollar.
- Is the lottery truly unbeatable?
- Yes. These two fixed-prize games both have negative EV, meaning that one is guaranteed to lose money in the long term.
- Is there a way to improve expected value of winning?
- No. There is no way to game them by structuring bet. There is no strategy element.
In short, in this case, there is no free money lying on the ground. In the next post we will continue our journey and investigate the last game in town - TOTO. Stay tuned for a dive into the deep end of Singapore Pools.