# Introduction

Anyone who is mildly excited by the idea of becoming a millionaire overnight would surely know TOTO: one of Singapore’s few biggest lotteries. I have to admit that I disagreed with participation in lotteries prior to writing this post. In fact, I wanted to write a post to show that the odds were stacked against players. My view was that TOTO wasn’t worth the money. It worsens inequality because it’s akin to having people pool money and letting a computer randomly select an accidental millionaire. It is a gamble with huge variance: you could win millions or lose hundreds, or possibly tens of thousands if you played big and frequently enough. It appeared to me a losing bet, because the probability of winning anything is approximately 1.86%.

This prejudice was unseated after some exploratory analysis and simulation. In this post, I show that TOTO is fairer than it seems, and I propose a way to maximise your expected winnings.

# How to Play TOTO

Most of the content in this section was taken from Wikipedia. I’ve reproduced it here for convenience.

## Picking Numbers

Players pick at least six numbers that have a value between 1 and 49. They can do this through several types of bets:

1. QuickPick: The computer randomly selects six numbers.
2. Ordinary Bet: The player selects six numbers of his/her choosing.
3. System 7 to System 12: The player selects seven to 12 numbers.
4. System Roll: The player selects five numbers. The sixth number is a guaranteed winning number.

The cost of each type of bet is as follows:

Bet Type Cost
Ordinary \$1
System 7 \$7
System 8 \$28
System 9 \$84
System 10 \$210
System 11 \$462
System 12 \$924
System Roll \$44

Note that the cost of the bet is exactly proportional to the number of combinations that bet contains. For example, a System 7 bet gives you seven numbers, from which we can make seven six-number combinations. Therefore, we can think of the cost as simply \$1 per six-number combination.

## Winning Prizes

The computer selects six numbers plus an additional number as the winning combination. There are a total of seven prize groups, which are defined as such:

Prize Group Matches Prize Odds of Winning Probability
1 6 numbers 38% of prize pool 1 in 13,983,816 0.0000071%
2 5 numbers + additional number 8% of prize pool 1 in 2,330,636 0.000043%
3 5 numbers 5.5% of prize pool 1 in 55,491 0.0018%
4 4 numbers + additional number 3% of prize pool 1 in 22,197 0.0045%
5 4 numbers \$50 per winning combination 1 in 1,083 0.092%
6 3 numbers + additional number \$25 per winning combination 1 in 812 0.12%
7 3 numbers \$10 per winning combination 1 in 61 1.64%

# The Data

Historical data on TOTO is available on Lottolyzer. You can download a CSV file that contains the winning numbers from July 2008 and payouts from June 2011. The data I use in this post were from October 2016 onwards, when the rules described above were implemented.

The data does not provide the total sales per game. However, we can infer this easily using PG 4, because this is the prize pool that has always had at least one winner, and has not been affected by cascading. Cascading occurs when there are no winners in all higher PGs than the PG in question. In the case of PG 4, cascading would only occur if there were no winners in PG 1-3. Hence, we could argue that backward induction using the PG 4 prize pool is fairly accurate, assuming that the reported winnings are true.

## Fun Fact: An Equal Share for Singapore Pools and Players

Before we dive into the math of TOTO, we look at how the pie is split between Singapore Pools (SG Pools), players, and the government. We take it that SG Pools and the government are separate entities because SG Pools has the word “Private” in its name despite being a subsidiary of the Tote Board.

### Players and SG Pools

I’m sure many would assume that SG Pools makes money from each game. Naturally, because of the affiliation between SG Pools and the government, I’m also sure many would question why only 54.5% of the total sales are reserved for Prize Groups (PG) 1-4, leaving a large proportion of 45.5% for the small prize winners. However, what is not as obvious is that there is no limit on the prize pools for PGs 5-7. The individual prizes are fixed, but the number of winners is not. Therefore, SG Pools only makes money if the total payout for PGs 5-7 is less than 45.5% of total sales.

Accounting for all PGs, the data suggests that the spoils from TOTO have been shared almost equally between SG Pools and players! See the graphs below on the total earnings (sales minus payouts) by SG Pools and players from TOTO games. SG Pools has made approximately nothing on average. While its profitable games were centered at about \$1.5 million to \$1.75 million, it has had games with massive losses that offset these gains. It’s earnings have also had a high standard deviation of \$2.1 million. Overall, from October 2016 to today, SG Pools has made a loss of \$19,000., Note that this is small relative to the size of each game.

Therefore, on average, SG Pools does not benefit at the expense of players. We could say that the game is fair, but we will assess this with some statistical rigour in the sections to come.

### The Government

Oh yes. We all love taking a peek into the government’s pockets. The government gains from TOTO because of the deduction of GST from the total sales of TOTO tickets. Thus far, we have been working with tax-deducted total sales, and we can easily calculate the total tax revenue from these figures. We should note that the taxes collected are always positive, and that the government takes no risk in each game. That is an incredibly sweet deal.