THE MATH IN TOTO LOTTERY AND A GOOD TIP

My previous blog showed for the 6 months Jan-Jun 2024 Singapore Toto provided an average chance of hitting the jackpot of 1 in 7,239,221 compared to probability theory of 1 in 13,983,816. All gaming systems are based on probability theories so how can Toto throw up outcomes of winners twice the expectation? One would be forgiven to ask if there were phantom winners.

This blog looks into the probabilities of winning under the different systems, ie systems 6, 7, 8, 9, 10, 11, 12 and system roll.

System 6 (S$1):

Singapore Toto is picking 6 winning numbers our of 49 numbers. The total number of combinations of 6 numbers as showED in the computation in the previous blog is 13,983,816. If a punter buys 1 bet, the probability of hitting jackpot is 1 in 13,983,816.

System 7: (S$7):

The number of combinations of 7 numbers out of 49 numbers is :


The number of winning combinations of 6 out of 6 is:


The number of loosing combinations of 1 out of remaining non-winning numbers is :


The probability of picking 7 numbers which include the 6 winning numbers is :


5.0057867e-7 is scientific expression for huge numbers which means move the decimal places 7 points to the left.
P = 0.0000005005867

The probability of hitting jackpot with a system 7/49 is = 0.00000050057867 or 0.000050057867 %

Expressed as a fraction P = 1/0.00000050057867 = 1,997,688
The probability of hitting jackpot with a system 7/49 bet is 1 in 1,997,688

Compare this to buying 7 x 6/49 bets which has a probability of 7/13,983,816 = 5.0057867e-7 which is 1 in 1,997,688

Thus it does not matter whether a punter buys 1 x system 7/49 bet for $7 or 7 system 6/49 bets @ $1 for S$7, the probability of hitting jackpot remains the same at 1 in 1,997,688

System 8: (S$28):

The number of combinations of 8 out of 49 numbers is :


The number of winning combination of 6 out of 6 numbers is:


The number of loosing combinations of 2 out of remaining non-winning numbers is:


The probability of picking 8 numbers which include the 6 winning numbers is:


The probability of hitting jackpot with a system 8/49 is 0.000002002314675765185 or 0.0002002314675765185 %.

Expressed as a fraction P = 1/0.000002002314675765185 = 499,422

The probability of hitting jackpot with a system 8/49 bet is 1 in 499,422

Compare this to buying 8 x 6/49 bets which has a probability of 28 / 13,983,816 = 2.002314675765185e-6 which is 1 in 449,422    

Thus it does not matter whether a punter buys 1 x system 8/49 bet for $28 or 28 system 6/49 bets @ $1 for S$28, the probability of hitting jackpot remains the same at 1 in 499,422

System 9 ($84), System 10 ($210, System 11 ($462) and System 12 (924):

The same computation will show the probability is the same for :
1 System 9/49   ($84)   or 84 system  6/49 bets @ $1 the probability is 1 in 166,474
1 System 10/49 ($210) or 210 system 6/49 bets @ $1 the probability is 1 in 66,590
1 System 11/49 ($462) or 462 system 6/49 bets @ $1 the probability is 1 in 30,268
1 System 12/49 ($924) or 924 system 6/49 bets @ $1 the probability is 1 in 15,134

The House never loses:

As far as probability theory goes, if you buy the following number of bets, you should be able to strike the jackpot but it will cost you ::

System 6  : $1    x 13,983,816 bets = $13,983,816
System 7  : $7     x 1,997,688 bets   = $13,983,816
System 8  : $28   x 99,422      bets   = $13,983,816
System 9  : $84   x 166,474    bets   = $13,983,816
System 10 : $210 x 66,590     bets   = $13,983,900
System 11 : $464 x 30,268     bets   = $13,983,816

As can be seen here, gaming systems are based on probability theory. It all boils down to the probability of picking 6 numbers out of 49 which is 1 in 18,983,815 chance. Punters will never win by going all out because :

1. The guaranteed price money is way too low. 

2. Only 38% of pool money is allocated to jackpot prize. It may not be enough. 

3. The sales may not be enough. As shown in the table in the previous blog, for the 6 months Jan-Jun 2024, ticket sales cannot support a $13,983,816 payout. 

On top of these numbers, there is no guarantee a punter can hit jackpot at his attempt at all-out bets. There is a phenomenon called geometric distribution or waiting time distribution. This describes the probability of observing the first "success" after a certain number of "failures". For example, in a throw of coins, if 'head' is the winner, there is a 50% chance of success. However, it may take a few failures before a success turns up. There could be a run or streak of tails before a head turns up. Probability theory is true if the events are played over many times. The more the times played, the more it's true there is 1 in 13,983,816 chance of hitting jackpot with a 6/49 bet.

Systems roll ($44):

A systems roll bet is where a punter picks only 5 out of the 6 winning numbers. In other words, it is getting exactly 5 correct numbers out of 6 winning numbers in a lottery where you choose from 49 numbers

The number of combinations of 6 numbers out of 49 numbers is :

The number of combinations of 5 out of 6 is:

The number of combination of 1 out of remaining numbers is:

The number of combinations of favourable outcomes is:

The probability of picking 5 correct numbers is:

The probability of hitting jackpot with a system roll is 0.0018499 %

Expressed as a fraction P = 1/0.000018499 =  1 in 54,201

Compare this to buying 44 x 6/49 bets @ $1 which has a probability of 44 / 13,983,816 = 0.0000031464945 or 0.000464945 % or 1 in 317,814 chance.

There seems to be an anomaly here. A punter is better off putting his $44 on a system roll bet where he has a 1 in 54,201 chance of hitting jackpot than to buy 44 x 6/49 bets of @ $1 where his chance is 1 in 317,814.  One system roll gives him roughly 6 times more chances than 44 bets of 6/49.

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