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.

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

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

**QuickPick:**The computer randomly selects six numbers.**Ordinary Bet:**The player selects six numbers of his/her choosing.**System 7 to System 12:**The player selects seven to 12 numbers.**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.

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% |

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.