Singapore TOTO

In the Singapore game of TOTO, 6 numbers plus one additional number are drawn at random from the numbers 1 to 45. In the Ordinary game, players spend 50c per game (minimum 2 games) and they choose 6 numbers in the hope of becoming instant millionaires.

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Combinations (for background on this section)
Probability and Poker

A prize pool is established at 55% of sales for a draw. Typically, $2.8 million dollars is "invested" in each game - and games are offered twice per week. This is quite a lot for a country of 4.5 million people...

Plenty of other countries have similar Toto games, usually called Lotto. The more numbers in a game, the worse your chances become.

Summary of the Prizes

Grp	Prize Amount	Winning Numbers Matched
1	33% of prize pool (min $500K)	6 numbers
2	13% of prize pool	5 numbers + additional number
3	13% of prize pool	5 numbers
4	13% of prize pool	4 numbers + additional number
5	$30 per winning combination	4 numbers
6	$20 per winning combination	3 numbers + additional number

TOTO 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 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 1 in C(45,6). That is:

That is, there are 8,145,060 ways of choosing 6 numbers from 45 numbers but there is only one correct combination.

So there is 1 chance in 8,145,060 of getting the Group 1 prize.

This means we have to buy 8 million tickets 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 39 remaining numbers that did not win anything [C(39,1)].

There is 1 chance in 39 that we chose the additional number, so multiply by 1/39.

So there is 1 chance in 1,357,510 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 39 remaining numbers that did not win.

So there is 1 chance in 34,808 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 39 remaining numbers that did not win anything [C(39,2)]. But we chose 6 numbers originally, so there are 2 chances in 39 that we chose the additional number, so multiply by 2/39.

So there is 1 chance in 14,290 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 39 remaining numbers that did not win.

So there is 1 chance in 733 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 the 39 remaining numbers that did not win anything [C(39,3)]. But we chose 6 numbers originally so there are 3 chances in 39 that we chose the additional number, so multiply by 3/39.

So there is 1 chance in 579 of getting the Group 6 prize.

Reader's Question

A reader wrote in to ask:

"In that group 6 (3 + additional number), there are 579 combinations, please help me to know all the possible 579 combinations."

My reply

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:

= 1.2277 × 10^-7 + 7.3664 × 10^-7 + 2.8729 × 10^-5 + 6.9979 × 10^-5 + 1.3646 × 10^-3 + 1.7262 × 10^-3

= 3.1904 × 10^-3

So the odds of getting any prize:

= 1 in 313.44.

Which are the Best Toto Numbers?

Click here for the Toto number frequency chart (at Singapore Pools site.)

Of course, there are no "best" numbers, but it is interesting to see which ones are most frequent.

An Interesting Problem...

On Monday 11 June 2001, the winning numbers for Singapore Toto were 1,10,19,23,29,45 with additional number 34.

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