# How to Predict and Win Singapore TOTO with Mathematics

In the Singapore game of TOTO, 6 numbers plus one “additional” number are drawn at random from the numbers 1 to 49. In the Ordinary game, players spend $1 and they choose 6 numbers in the hope of becoming instant millionaires. Yet, what are the odds of winning the lucky numbers for Singapore TOTO? Let Tropika Club investigate if we can predict and win Singapore TOTO with mathematics. This article is adapted from Interactive Mathematics (source).

## What is Singapore TOTO?

As explained in the Singapore Pools website, to participate in the TOTO Game, the Participant has to either place a Bet through the Account Betting System or buy a Ticket at the Outlet. Details on how to place Bets and buy Tickets are set out in the TOTO Game Rules (Remote Betting) and TOTO Game Rules (Outlet Betting) respectively.

- The Stake payable for each set of six (6) numbers selected for every draw shall be one (1) dollar. The Stake for a Ticket and/or Bet shall be no less than one (1) dollar.
- Save as expressly provided in the Rules and/or the Account Opening and Operating Terms and Conditions, Stakes paid to the Company in respect of Bets shall not be refundable to the Participants.
- An intending Participant may participate in the TOTO Game by making an Ordinary Entry whereby the Participant purchases a set of six (6) numbers from a total of forty-nine (49) numbers available for selection in the Game.
- An intending Participant may participate in the Game by making a System Entry whereby the Participant purchases all possible combinations of sets of six (6) numbers from a selection of five (5) numbers with the sixth (6th) number from each of the remaining forty four (44) numbers of the total of forty-nine (49) numbers available for selection in the Game; and/or
- all possible combinations of sets of six (6) numbers from a selection of seven (7) to twelve (12) numbers out of the total of forty-nine (49) numbers available for selection in the Game.

An intending Participant may choose to make one or more of the abovementioned System Entries as follows:-

**System Roll for a selection of five (5) numbers as described in Rule 5.2(a)****System 7 for a selection of seven (7) numbers****System 8 for a selection of eight (8) numbers****System 9 for a selection of nine (9) numbers****System 10 for a selection of ten (10) numbers****System 11 for a selection of eleven (11) numbers****System 12 for a selection of twelve (12) numbers**

An intending Participant of any System Entry shall pay the requisite stake for the System Entry chosen as follows:

## How to Win at Toto?

There shall be a prize pool available for distribution of prizes which may be won for each draw (“Prize Pool”). The Prize Pool shall be an amount equal to fifty-four per cent (54%) of the net amount of total stake collection of that draw less the Goods and Services Tax thereon. Allocations shall be made to seven (7) groups of prizes based on the number of winning numbers matched in the following manner. There shall be a minimum guaranteed pool amount of $1,000,000 for the Jackpot Prize for each draw (“the Minimum Guaranteed Amount”). In the event that the Jackpot Prize amount plus any “snowballed” amount(s) (collectively, the “Prize Amount”) is less than the Minimum Guaranteed Amount, the Prize Amount shall be increased by the Company to the Minimum Guaranteed Amount, and paid out to and shared equally among the total number of winning shares of the Jackpot Prize.

## Let’s look at TOTO winning Odds.

Recall (from the Combinations section) that the number of ways in which *r* objects can be selected from a set of *n* objects, where repetition is **not** allowed, is given by:

We can write (and type) the left hand side more conveniently as *C*(*n*,*r*).

Now let’s look at the probabilities for each prize.

### Group 1 (Choose all 6)

The odds of winning the top Group 1 prize are *C* (49,6)). That is:

That is, there are 13,983,816 ways of choosing 6 numbers from 49 numbers but there is only one correct combination.

So there is **1 chance in 13,983,816** of getting the Group 1 prize.

This means we have to buy almost 14 million tickets (at a cost of $14 million) before we can confidently say we will probably win the top prize…

### Group 2 (5 + additional)

Odds:

**Explanation: **We chose 5 of the 6 winning numbers [*C*(6,5)], and chose the correct “additional” number from the \displaystyle{43}43 remaining numbers that did not win anything [*C*(43,1)].

There is 1 change in 43 that we chose the additional number, so multiply by 1 / 43.

So there is **1 chance in 2,330,636** of getting the Group 2 prize.

### Group 3 (5 correct)

Odds:

We chose 5 of the 6 winning numbers and chose 1 number from the 43 remaining numbers that did **not **win. In the Group 3 prize, we **cannot** include the “additional” number, so we need to multiply by the probability of the remaining43 numbers **not** containing the additional number which is 1 – 1/43 = 42/43 .

So there is **1 chance in 55,491** of getting the Group 3 prize.

### Group 4 (4 + additional)

Odds:

We chose 4 of the 6 winning numbers [*C*(6,4)], and chose 2 numbers from the 43 remaining numbers that did not win anything [*C*(43,2)]. But we chose 6 numbers originally, so there are2 chances in 43 that we chose the additional number, so multiply by 2/43.

So there is **1 chance in 22,197** of getting the Group 4 prize.

### Group 5 (4 correct)

Odds:

We chose 4 of the 6 winning numbers and chose 2 numbers from the 43 remaining numbers that did not win. Once again, we need to consider the probability of the additional number not being one of our 2 remaining (non-winning) numbers. This probability is 1 – 2/43 = 41/43. So we multiply by 41/43.

So there is **1 chance in 1,083** of getting the Group 5 prize.

### Group 6 (3 + additional)

Odds:

We chose 3 of the 6 winning numbers [*C*(6,3)], and choose 3 numbers from the43 remaining numbers that did not win anything [*C*(43,3)]. But we chose 6 numbers originally so there are3 chances in 43 that we chose the additional number, so multiply by 3/43.

So there is **1 chance in 812** of getting the Group 6 prize.

### Group 7 (3 correct)

Odds:

We chose 3 of the 6 winning numbers and chose 3 numbers from the 43 remaining numbers that did not win. Again, we need to consider the probability of the additional number not being one of our 3 remaining (non-winning) numbers. This probability is1 – 3/43 = 40/43. So we multiply by 40/43.

So there is **1 chance in 61** of getting the Group 7 prize.

### Odds of Getting Any Prize:

Total probability of getting **any** prize is simply the sum of all the probabilities for Group 1 to Group 6 prizes:

## System Entries

In most Lotto and Toto games, you can buy a “System”. Your chances of winning increase, but of course, you pay more as well. For example:

**System 7** means you choose 7 numbers (instead of the usual 6). This gives you 7 times the chance of winning (so it costs 7 times as much), since it is equivalent to buying 7 different 6-number games, or *C*(7,6). Say you chose 1, 3, 5, 7, 9, 11, 13 as your 7 numbers. You have the following 7 ways of winning if the 6 winning numbers happened to be:

1 3 5 7 9 11

3 5 7 9 11 13

1 5 7 9 11 13

1 3 7 9 11 13

1 3 5 9 11 13

1 3 5 7 11 13

1 3 5 7 9 13

**System 8** means you choose 8 numbers and it gives you the equivalent of 28 ordinary bet combinations, so costs 28 times as much, or *C*(8,6).

Similarly, **System 9** gives you *C*(9,6)=84 ordinary bet combinations, **System 10** gives *C*(10,6)=210 ordinary combinations, **System 11** gives *C*(11,6)=462 combinations and **System 12** (the maximum in the Singapore game) gives \*C*(12,6)=924 combinations. The probability of winning with a System 12 is 924 times the probability of winning when you buy 1 game, that is:

## Changes of Winning TOTO Group 6 with System 12

Reader David wrote in to ask about the probability of winning a Group 6 prize when you buy a System 12. He asked what happens when the probability appears to be greater than 1. What he means is this: For a Group 6 prize, the probability is 1 in 812, so (it appears) a System 12 will give you 924 chances in 812 or 1.138 chances – greater than 1! However, this refers to the high chance in relation to your 12 chosen numbers – not to the total 49 possible numbers.

It just means you can be “almost certain” to get a Group 6 prize when you buy a System 12.

In fact, you would expect to win 924 / 321 = 2.88 (almost 3) prizes of some sort.

If you win a Group 6 prize with a System 12, you actually get 28 Group 6 shares (28 × $25 = $700) and because you have 3 correct numbers as well, you get 56 Group 7 shares (56 × $10 = $560) so your total prize amount is $1,260 (or a “profit” of $1260 – $924 = $336). Don’t believe anyone who says any strategy is a “sure thing”. The only sure thing is the “house” (or in many cases, the government) are the real winners.

## Which are the “Best” TOTO Numbers?

Of course, there are no “best” numbers, but it’s interesting to see which ones occur most frequently.

The number with highest frequency (40, at 101 times) has come out 1.7 times as often as the lowest frequency number (45, at 58 times).

The data for the above chart comes from the Toto number frequency chart (at Singapore Pools site.)

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