NOTE: At the time, there were only 45 numbers to choose from in Singapore Toto, not 49 as there is now.

Let us start with a similar but simpler game. Say we have 9 numbers (1,2,3,4,5,6,7,8,9) and we draw 4 numbers each time. Four numbers have been drawn (say 1,2,3,4) We want to find the probability that two of these four numbers come out again in the next draw.

We divide the numbers into 2 groups:

Category A: 1,2,3,4 (four numbers came out in the first draw)

Category B: 5,6,7,8,9 (five remaining numbers)

Suppose 1 and 2 repeat:

1,2,5,6

1,2,5,7

1,2,5,8

1,2,5,9

1,2,6,7

1,2,6,8

1,2,6,9

1,2,7,8

1,2,7,9

1,2,8,9

There are C(5,2) = 10 numbers in this group. We choose 2 from the 5 numbers from Category B.

Similarly there is another group in which 1,3 repeat, and another where 1,4 repeat, then 2,3 repeat etc.

Altogether there are C(4,2) = 6 groups (4 is the number of numbers in group A and 2 of them are drawn).

So the probability of the 2 numbers repeating in 2 different draws is:

`(C(5,2)xxC(4,2))/(C(9,4))=(10xx6)/126=10/21`

The TOTO case is similar: In each game, we draw 7 numbers (6 plus one additional) from a pool of 45. We want to find the probability of having 5 numbers repeating.

There are 7 numbers in Category A and 38 numbers in Category B.

The number of ways to have the first five numbers repeating:

We choose 2 from 38 numbers in group B: C(38,2) = 703.

There are C(7,5) = 21 groups, and there are 7 numbers in group A and 5 of them are drawn.

The denominator is C(45,7) = 45 379 620.

The probability is

`(703xx21)/(45\ 379\ 620)=(4\ 921)/(15\ 126\ 540)`

`=3.2532xx10^-4` or `1` in `3073.9`.

Of course, this is only the probability of having EXACTLY 5 numbers repeating.

The probability of 6 numbers repeating can be found using similar working:

`(38xx7)/(45\ 379\ 620)=133/(22\ 689\ 810)`

`=5.8617xx10^-6` or `1` in `170\ 600`.

Also the probability of having all the same 7 numbers coming out again is `1/(45\ 379\ 620)`.

The answer for the probability of having AT LEAST 5 repeating numbers in the next draw is

`(14\ 763)/(45\ 379\ 620)` `+(266)/(45\ 379\ 620)` `+1/(45\ 379\ 620)`

`=167/(504\ 218)`

`=0.00033121`

or 1 in 3019.2.

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