## Introduction to TOTO

Lottery is part and parcel of many Singaporean families. Being “exposed to gambling” from young — us cousins in the family used to hang around our grandparents’ place every weekends and “tikam tikam” or guess numbers for their weekly bets. We picked individual numbers out of a jar for TOTO and 4D, sometimes we win most of the times we didn’t.

Most writings I encountered so far don’t go beyond the obvious — cost and odds of bet types or prize amounts. Playing the Devil’s Advocate, I thought it would be interesting to offer some of my own thoughts on top of the good work by predecessors on why higher bet types are prohibitive yet offer the best chance of positive returns over a long run.

## Disclaimer

This post in no way encourages gambling. Betting is still a game of luck and probabilities. TOTO is a gamble with huge variance. You could lose hundreds, or thousands of dollars if you make many huge and frequent bets. It is a sure-lose way of throwing away money as the probability of winning anything is 1.85%. Even going for exams without studying will get you more than average 2 points overall.

However, is this prejudice without looking much more in-depth into the gamble itself? The point of getting into a gamble anyhow, is to opt for the bet with the highest probabilities based on the maximum risk that you can take. Predecessors have unseated this prejudice as well with exploratory analysis and simulation.

TOTO is fairer than it seems: there is a way to maximise expected winnings with enough capital. And I don’t mean buying up all of the different combinations. |

## How to win TOTO

Simple enough, 6 (min) to 12 (max) numbers are picked between 1 — 49, through the various bet types:

**QuickPick:**The system generates 6 random numbers between 1 — 49.**Ordinary Bet:**Select 6 numbers between 1 — 49.**System 7 to System 12:**Select up to 12 numbers between 1 — 49. More numbers gives you more Ordinary Entries in one bet.**iTOTO:**The system generates 12 random numbers between 1 — 49. It is a Quick Pick System 12 Entry divided into 28 units: Unit cost and prize amount is divided by 28.**System Roll:**Select 5 numbers between 1 — 49. The remaining rolling number ‘R’ represents all remaining 44 numbers.

Bet type | Number of Ordinary entries | Cost |

Quickpick / Ordinary | 1 | $1 |

System 7 | 7 | $7 |

System 8 | 28 | $28 |

System 9 | 84 | $84 |

System 10 | 210 | $210 |

System 11 | 462 | $462 |

System 12 | 924 | $924 |

iTOTO | 924 | $33 |

System Roll | 44 | $44 |

The prize amounts for the various prize groups are derived from Singapore Pools website.

## How hard is it to win a TOTO prize?

Generally, betters are highly interested in efforts required to reap the rewards versus the costs of attaining it. A higher bet type will have more ‘ordinary’ chances, thus better odds and higher probability. It’s all co-related.

Odds and probability for Ordinary bet are mathematically derived. For System 7 to 12, they are derived from Monte Carlo simulation . 50 million TOTO games was simulated to figure out the probabilities of achieving matching combinations with each bet type.

A bettor has a **1-in-54** chance of winning any prize for an Ordinary bet and down to a **1-in-7** chance for System 12 bet over the long run.

**Strategies for winning TOTO**

In short, there is no sure-win strategy. Ignore all the high-low, odd-even, sum-of-numbers, frequency-of-numbers strategies out there. Each draw is independent from the next draw; any number drawn is independent from each other as well. Lucky numbers, popular numbers etc., they do not mean anything really. Trends arise after a long run and can be explained by statistics.

Taking data for the **past 1254 games (Game 2341 to 3594)**, we have come up with some statistics to examine.

**Odd/Even numbers combinations**

Each number has an equal 1/49 chance of being drawn from the start. Similarly with 25 odd and 24 even numbers, there’s an almost equal chance of an odd/even number drawn with a very slight probability that odd number has a higher chance of being drawn.

Therefore we observe that having equal number of odd/even numbers drawn is the highest (37.6%). Having a odd/even combination of 2-4, 3-3 and 4-2 becomes statistically higher at 81%.

Without going into the high-low range or high-mid-low or even each number range grouping (1-9, 10-19, 20-29 etc), we find that it will statistically tend towards equal distribution.

**Same number appear from the last draw** (not same combination of numbers)

At the end of 1 draw (6 tries), the probability of a particular number being drawn is 0.1292 [1/49 x 1/48 x 1/47 x 1/46 x 1/45 x 1/44]. The probability of the particular number being drawn at the next draw as well is 0.01669 [0.1292 x 0.1292]. Since there are 49 numbers to draw, the probability of any number being drawn in the next draw is much higher.

We observe that having 0 or 1 number appearing from the last draw is roughly the same. Having 0 or 1 number appearing from last draw is statistically high at 85%. In reality, we know as well that it is more improbable for 2 or more same numbers to appear from the previous draw.

Similarly, one can plot the same numbers that had appeared within the past 2, 3 or 4 draws to notice the trend with enough data.

Is the set of (1,2,3,4,5 & 6) impossible to take place? Out of 13,983,816 combinations, all numbers or even combinations has equal chances of appearing; we just haven’t had enough actual games yet.

Here, we are more interested in examining results of probabilities stimulations.

## Is TOTO a fair gamble?

A fair gamble is one where you have a 50-50 chance of winning $1 with an expected payoff of zero. If you were to gamble in countless numbers of fair gamble, you make nothing in the long run.

Expected Payoff = (50%×$1) + (50%×−$1) = $0 |

A better-than-fair gamble is one where you have a higher chance of winning $1 with an positive expected payoff (**due to higher probability**). Say a 55-45 chance of winning $1. It could also be one whereby you win more dollars than you lose in a rarer chance, leading to a positive expected payoff as well (**due to higher payoff**). Over the long run, you will earn money.

Expected Payoff ( = (55%×$1) + (45%×−$1) = $0.10due to higher probability) |

Expected Payoff (due to higher payoff) = (40%×$2) + (60%×−$1) = $0.20 |

TOTO is a better-than-fair gamble, regardless of the individuals’ bet type or prize pool size.

HUH? Now by this time, people will be scratching their heads. How can TOTO be a fair gamble? It is **1 out of 13,983,816** to win the jackpot and **1 out of 54** to win any group prizes with an ordinary bet. With the most expensive System 12 bet, the odds are brought down to **1 out of 17,857** for the jackpot and **1 out of 7** for any group prizes. No way is this 50-50 or even better, and everyone is bound to lose over the long run.

I thought it will be interesting to look at the overall picture and then delve deeper.

## Is Singapore Pools making more money than it does giving out?

Just like a casino, we would expect that Singapore Pools is making much MORE money than it is giving out if TOTO was not a fair gamble. Is Singapore Pools making a huge profit all this while?

As a background, 54.5% of TOTO sales in each draw is for distribution of prizes in the Groups 1 — 4 which may be won for each draw. The rest of the TOTO sales is reserved for Groups 5 — 7.

Prize group | Numbers matched | Prize amount |

1 | 6 nums | 38% of prize pool (Minimum Guaranteed of $1,000,000) |

2 | 5 nums + 1 add | 8% of prize pool |

3 | 5 nums | 5.5% of prize pool |

4 | 4 nums + 1 add | 3% of prize pool |

Total | 54.5% of prize pool |

The prizes for the higher groups 1 — 4 are fixed as a percentage. The number of winners does not matter as the group prizes will be shared equally among the winners in the same prize group.

Prize group | Numbers matched | Prize amount |

5 | 4 nums | no. of winners X $50 |

6 | 3 nums + 1 add | no. of winners X $25 |

7 | 3 nums | no. of winners X $10 |

Total | 45.5% of prize pool |

While the prizes for the lower groups 5 — 7 are fixed dollar amounts, the number of winners does matters as all winners get the same dollar amounts in the same prize group.

Singapore Pools **makes money only **if total payout for the lower groups 5 — 7 is **less than 45.5% **of total sales.

Interestingly, Singapore Pools has made almost nothing when looking at data from October 2016 onwards. The overall sales is negated by overall winnings from the betters. From this simple analysis, we can infer that TOTO is a fair gamble. By fair, we do not mean that the game is fair to you but as a single party (betters) to single party (Singapore Pools) comparison, both parties have an almost equal win in absolute dollars.

## Bigger Prize Pool translates to Bigger net expected value

Prize Groups 1 — 4 are **allocated 54.5%** of the prize pool or total sales of TOTO. Generally, the larger the prize pool, the larger the winnings. All bet types encountered a positive net expected value but over the long run. In other words, a better-than-fair gamble regardless of the bet types.

The only issue is that people naturally “give up” after not winning for some time. We don’t see players participating in all TOTO games over a prolonged period of time.

The compulsive gambler too does not follow a religious system in their betting — they bet however big they want or “tikam tikam” over a few different bet types. They upped their bets after winning something, which sometimes turn their winnings into bigger losses instead.

Hence, I would agree that short-term players end up contributing to the prize pool for other lucky or determined players to win. These said players do not play enough to break even or profit through a massive win. It would take thousands of games and possibly thousands of dollars in costs to win the big Prize Groups 1 — 4. Even in 10 years of our lifetime, only 1040 games are played.

## Simulation of 1040 games

To cut the long story short, predecessor had done 1,000 simulations of 1,040 TOTO games each for each bet type to identify possible outcomes.

The outcome could be summarised in the below table

The outcome range shows the maximum and minimum net balances from the simulations; you should be willing to be down as low as the minimum balance and expect at most, the maximum balance during a 10-year TOTO regime.

By betting on only System 9 for 10 years, I should be prepared both mind and soul to lose as much as $75,076 and make at most $4.0 million. As the median, I would expect to earn approximately $175,359 over ups and downs.

System 7 and System 8 turned out to have negative median outcomes instead, largely due to the low probability of hitting any big wins.

The above simulations were strict in a few criteria: no snowballing or cascading of prize monies, no sharing of prizes. In reality, these few factors do change the flavor in the expectations further.

## A word of caution

Statistics however do not determine trends. Take a coin with Head/Tail sides. By taking 1 million simulations for example, we find that the outcome is roughly 50% for both Head/Tail. This does not determine however that one side cannot happen for a long time before the other side happens e.g. 20 Heads appear before 1 Tail.

A sample coin flip: THHHHHHHHHHHHHHHHHHHHTTTHTHT |

The simulations for TOTO above is carried out for thousands of games. Over 10 years, only 1040 actual games are carried out — not quite a lot of games if you were to ask me. There is nothing against stopping you from losing tens of games before winning a small Prize Group, or even hundreds of games before winning a large Prize Group.

Yet, it does not change the fact that TOTO could be a better-than-fair gamble as the rare payoffs are sufficiently large (to compensate for the probability of winning them) and the smaller payoffs are sufficiently common. It also depends on the bet type you intend to stick to (for a long time) and your time frame.

Over the short run, TOTO is a losing bet for most of us who do not stick around for enough games to land a large win. Over the long run, TOTO becomes a winning bet, though it will take a very long time to realise a net positive expected value. Statistics is in your face to either gamble on the more expensive bet types (System 10-12) for a very long time or not bother at all.

It sounds ridiculous but rational at the same time. The only thing holding us back is our available capital. That is why you do not hear of many people even playing System 10 consistently over a long period of time.

Could we have it any worse?

Credits to the writer of TOTO: An Irrational, Rational Approach to Singapore’s Second Biggest Lottery for stimulating my thoughts.