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:
$C_{r}=r!(n−r)!n! $
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 $1$ in C(49,6). That is:
$C(,)1 =,,1 $
$=7.15×_{−8}$
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:
$C(,)C(,)×C(,) ×431 $
$=,,258 ×431 $
$=,,1 $
$=4.29×_{−7}$
Explanation: We chose 5 of the 6 winning numbers [C(6,5)], and chose the correct “additional” number from the $43$ remaining numbers that did not win anything [C(43,1)].
There is $1$ chance in $43$ that we chose the additional number, so multiply by $431 $.
So there is 1 chance in 2,330,636 of getting the Group 2 prize.
Group 3 (5 correct)
Odds:
$C(,)C(,)×C(,) ×4342 $
$=,,258 ×4342 $
$=,1 $
$=1.80×_{−5}$
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 remaining $43$ numbers not containing the additional number, which is $1−431 =4342 $.
So there is 1 chance in 55,491 of getting the Group 3 prize.
Group 4 (4 + additional)
Odds:
$C(,)C(,)×C(,) ×432 $
$=,,, ×432 $
$=,1 $
$=4.505×_{−5}$
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 are $2$ chances in $43$ that we chose the additional number, so multiply by $432 $.
So there is 1 chance in 22,197 of getting the Group 4 prize.
Group 5 (4 correct)
Odds:
$C(,)C(,)×C(,) ×4341 $
$=,,, ×4341 $
$=1082.75771 $
$=9.236×_{−4}$
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 (nonwinning) numbers. This probability is $1−432 =4341 $. So we multiply by $4341 $.
So there is 1 chance in 1,083 of getting the Group 5 prize.
Group 6 (3 + additional)
Odds:
$C(,)C(,)×C(,) ×433 $
$=,,, ×433 $
$=812.0681 $
$=1.23142×_{−3}$
We chose $3$ of the $6$ winning numbers [C(6,3)], and choose $3$ numbers from the $43$ remaining numbers that did not win anything [C(43,3)]. But we chose $6$ numbers originally so there are $3$ chances in $43$ that we chose the additional number, so multiply by $433 $.
So there is 1 chance in 812 of getting the Group 6 prize.
Group 7 (3 correct)
Odds:
$C(,)C(,)×C(,) ×4340 $
$=,,, ×4340 $
$=60.9051 $
$=1.642×_{−2}$
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 (nonwinning) numbers. This probability is $1−433 =4340 $. So we multiply by $4340 $.
So there is 1 chance in 61 of getting the Group 7 prize.
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 6number 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 $C(,)$.
Similarly, System 9 gives you $C(,)=84$ ordinary bet combinations, System 10 gives $C(,)=210$ ordinary combinations, System 11 gives $C(,)=462$ combinations and System 12 (the maximum in the Singapore game) gives $C(,)=924$ combinations.
The probability of winning with a System 12 is $924$ times the probability of winning when you buy 1 game, that is:
$,,924 $ or $1$ in $15,134$.
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.
A prize pool is established at 54% 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 5.5 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 (Singapore Toto)
Grp

Prize Amount

Winning Numbers Matched

1

38% of prize pool (min $1 M)

6 numbers

2

8% of prize pool

5 numbers + additional number

3

5.5% of prize pool

5 numbers

4

3% of prize pool

4 numbers + additional number

5

$50 per winning combination

4 numbers

6

$25 per winning combination

3 numbers + additional number

7

$10 per winning combination

3 numbers

Are you into buying 4D? If you have been an ardent fan of this relatively harmless form of gambling, I’m sure you must have thought about the odds of winning the first price many times haven’t you?
Well then, let me let you in on some statistics of winning something from Singapore Pools each time you put your money on 4 random numbers.
The statistics I’m about to broadcast may come as a shock to some of you, others maybe not so much. You see, I’ve asked those who purchase 4D around me before if they knew what they were getting themselves into monetary wise and they almost always replied, “Well, it’s all for that glimmer of hope to change our lives once and for all.”
It’s not a bad thing, really. I mean, we all resort to something in a bid to change our lives right? Putting a small amount of money in a gamble for larger winnings is just what they do. But the chances of even winning that consolation or starter prize though is just too low for me to ever consider joining in “just for the fun of it”.
Okay, here goes. According to Singapore Pools, the odds of winning any prize is 1 in 435. Not too bad right? Keep in mind this is for “any prize” and, I’m sure, not the prize you are looking to win. Following this, it is a 1 in 1,000 chance of winning a consolation prize or a starter prize.
Now comes the best part and possibly the part everyone is waiting for. The top 3 prizes. If you want to get anywhere near winning one of these, this is the kind of odds you are up against.
For all 3 top positions, you have a 1 in 10,000 chance of winning. Seriously, what are the odds of winning? Well, that’s “hope” for you right there.
To many people, this is just another pastime for them to wither some of their time away and give them something to look forward to. For others, it is a chance for them to hopefully change their lives around.
If you are still very sure that one fine day, you will emerge victorious, then please be my guest and continue doing what you need to do.
There is no right or wrong when activities are practiced in manageable doses. I’m hoping these statistics will have made some of you think twice the next time you are at a 4D booth and marking out those ‘winning numbers’ on that rectangular sheet of paper.
It can be a shot at winning some money but it should never be done to the extent of wrecking your life altogether.