The math behind TOTO: Here's why it (statistically) doesn't make financial sense to play

Traditions that are mainstays of Chinese New Year in Singapore include: Home visitations, Lou Hei, gambling with friends and family, and the TOTO Hong Bao Draw. This year, the Group 1 prize is $12 million and the draw will be held on 7 February 2020 at 9.30pm.

We agree that it might be fun to purchase "hope" (and we will probably be among those who buy a ticket ourselves), it is also true that Singaporeans will probably start their year better by abstaining and thus not losing money. Here's why.

RECAP OF THE TOTO RULES

For those who don't know how to play TOTO, here is quick summary of the key rules.

1. You pick at least 6 numbers between 1 and 49.
2. During each draw, 6 Winning Numbers and 1 Additional Numbers are drawn.
3. If your chosen numbers match at least 3 of the Winning Numbers, you win a prize. The amount you receive depends on how many of your chosen numbers match the Winning Numbers.
4. Another important caveat is that if is more than 1 winner in Groups 1 to 4, prizes will be divided equally among all winners of the group. The prize amounts for Groups 5 to 7 are fixed.

WHAT ARE THE ODDS OF WINNING THE TOTO JACKPOT (GROUP 1)?

When you play TOTO, you do so in hopes of winning the eye-watering top prize. But what are your odds of doing so? To answer this question, we dusted off our copy of Schaum's and refreshed our knowledge on Permutations and Combinations.

We need to match all 6 numbers from a possible list of 49. Since the numbers don't repeat, there is only 1 possible combination that will yield us the top prize.

Thus, C(49, 6) = 1/13,983,816

In other words, the odds of winning is about 1 in 14 million.

To say this is extremely poor odds would be an understatement.

Then again, we humans are bad at gauging immensely improbably events. One way to think about whether how much value your $1 lottery ticket has is to calculate its Net Present Value.

NET PRESENT VALUE OF A $1 TOTO TICKET

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The actual odds and prize value would can only be calculated once sales closes, since additional 54 per cent of sales in each draw would be added to grow the prize pool.

For the purposes of this calculation, we assume that the prize pool is $31,578,947 - based on $12 million being 38 per cent of the prize pool.

We also assume we only buy 1 TOTO ticket and ignore other players for now. In other words, this is how much your TOTO ticket is worth if you were the only one playing.

If we multiply the probability of winning by the prize money and sum up the total probable winnings, we will have a Present Value of $76.16. Subtracting the $1 we used to buy our TOTO ticket, we would get a Net Present Value of $75.16.

And if we add up all the probabilities of winning, the chances of actually winning something in any of the groups is 1.9 per cent.

Not bad, isn't it?

Unfortunately for you, Singapore Pools does not operate the draw just for you. When you consider other players and how they affect your winnings, your TOTO ticket's Net Present Value looks very different.

NET PRESENT VALUE OF A $1 TOTO TICKET (BASED ON HISTORICAL NUMBER OF WINNERS)

If you recall, the prize money in Group 1, 2, 3 & 4 will be split in the event that there are multiple winners for each group.

Again, it is impossible to calculate the expected value because there is no way to know for certain how many winners there would be in each category. We can, however, use the average of the number of winners from each group in the past 5 Hong Bao Draws.

If we take this into account and calculate the present value of our TOTO ticket, we get a very different conclusion:

The expected value has significantly reduced to $0.77. If we subtract the $1 used to buy our ticket, our Net Present Value is now negative $0.23.

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Now do you see why it doesn't make statistical sense to play TOTO?

Ironically, your TOTO ticket's Net Present Value would be highest at a time when the least players are playing the game.

All other players know that. But because none of the players have perfect information on when equilibrium is reached (where the Present Value of a $1 TOTO ticket is slightly above $1), it is impossible to "game" the system.

Thus, the real winner of every Hong Bao draw, remains Singapore Pools.

HAVE FUN RESPONSIBLY

On a lighter note, the participation in TOTO betting should be one of leisure and a topic with friends and family during this festive period. However, if you are thinking of getting rich just because of this Hong Bao draw, don't put too much hope into it as you would for any other regular TOTO draws.

Let's exercise a Responsible Play mentality to TOTO (or any other form of gambling) as a recreational activity and never as a get-rich-quick-scheme or form of "investment".

This article was first published in Dollars and Sense.