Introduction

Anyone who is mildly excited by the idea of becoming a millionaire overnight would surely know TOTO: one of Singapore’s few biggest lotteries. I have to admit that I disagreed with participation in lotteries prior to writing this post. In fact, I wanted to write a post to show that the odds were stacked against players. My view was that TOTO wasn’t worth the money. It worsens inequality because it’s akin to having people pool money and letting a computer randomly select an accidental millionaire. It is a gamble with huge variance: you could win millions or lose hundreds, or possibly tens of thousands if you played big and frequently enough. It appeared to me a losing bet, because the probability of winning anything is approximately 1.86%.

This prejudice was unseated after some exploratory analysis and simulation. In this post, I show that TOTO is fairer than it seems, and I propose a way to maximise your expected winnings.

How to Play TOTO

Most of the content in this section was taken from Wikipedia. I’ve reproduced it here for convenience.

Picking Numbers

Players pick at least six numbers that have a value between 1 and 49. They can do this through several types of bets:

  1. QuickPick: The computer randomly selects six numbers.
  2. Ordinary Bet: The player selects six numbers of his/her choosing.
  3. System 7 to System 12: The player selects seven to 12 numbers.
  4. System Roll: The player selects five numbers. The sixth number is a guaranteed winning number.

The cost of each type of bet is as follows:

Bet Type Cost
Ordinary $1
System 7 $7
System 8 $28
System 9 $84
System 10 $210
System 11 $462
System 12 $924
System Roll $44

Note that the cost of the bet is exactly proportional to the number of combinations that bet contains. For example, a System 7 bet gives you seven numbers, from which we can make seven six-number combinations. Therefore, we can think of the cost as simply $1 per six-number combination.

Winning Prizes

The computer selects six numbers plus an additional number as the winning combination. There are a total of seven prize groups, which are defined as such:

Prize Group Matches Prize Odds of Winning Probability
1 6 numbers 38% of prize pool 1 in 13,983,816 0.0000071%
2 5 numbers + additional number 8% of prize pool 1 in 2,330,636 0.000043%
3 5 numbers 5.5% of prize pool 1 in 55,491 0.0018%
4 4 numbers + additional number 3% of prize pool 1 in 22,197 0.0045%
5 4 numbers $50 per winning combination 1 in 1,083 0.092%
6 3 numbers + additional number $25 per winning combination 1 in 812 0.12%
7 3 numbers $10 per winning combination 1 in 61 1.64%

The Data

Historical data on TOTO is available on Lottolyzer. You can download a CSV file that contains the winning numbers from July 2008 and payouts from June 2011. The data I use in this post were from October 2016 onwards, when the rules described above were implemented.

The data does not provide the total sales per game. However, we can infer this easily using PG 4, because this is the prize pool that has always had at least one winner, and has not been affected by cascading. Cascading occurs when there are no winners in all higher PGs than the PG in question. In the case of PG 4, cascading would only occur if there were no winners in PG 1-3. Hence, we could argue that backward induction using the PG 4 prize pool is fairly accurate, assuming that the reported winnings are true.

Fun Fact: An Equal Share for Singapore Pools and Players

Before we dive into the math of TOTO, we look at how the pie is split between Singapore Pools (SG Pools), players, and the government. We take it that SG Pools and the government are separate entities because SG Pools has the word “Private” in its name despite being a subsidiary of the Tote Board.

Players and SG Pools

I’m sure many would assume that SG Pools makes money from each game. Naturally, because of the affiliation between SG Pools and the government, I’m also sure many would question why only 54.5% of the total sales are reserved for Prize Groups (PG) 1-4, leaving a large proportion of 45.5% for the small prize winners. However, what is not as obvious is that there is no limit on the prize pools for PGs 5-7. The individual prizes are fixed, but the number of winners is not. Therefore, SG Pools only makes money if the total payout for PGs 5-7 is less than 45.5% of total sales.

Accounting for all PGs, the data suggests that the spoils from TOTO have been shared almost equally between SG Pools and players! See the graphs below on the total earnings (sales minus payouts) by SG Pools and players from TOTO games. SG Pools has made approximately nothing on average. While its profitable games were centered at about $1.5 million to $1.75 million, it has had games with massive losses that offset these gains. It’s earnings have also had a high standard deviation of $2.1 million. Overall, from October 2016 to today, SG Pools has made a loss of $19,000., Note that this is small relative to the size of each game.

Therefore, on average, SG Pools does not benefit at the expense of players. We could say that the game is fair, but we will assess this with some statistical rigour in the sections to come.

The Government

Oh yes. We all love taking a peek into the government’s pockets. The government gains from TOTO because of the deduction of GST from the total sales of TOTO tickets. Thus far, we have been working with tax-deducted total sales, and we can easily calculate the total tax revenue from these figures. We should note that the taxes collected are always positive, and that the government takes no risk in each game. That is an incredibly sweet deal.

The taxes earned off TOTO have been both substantial and stable. Below are some statistics on the taxes earned from TOTO from October 2016 to today.

Statistic Figure
Average Taxes per Game $361,274
Std. Dev. of Taxes per Game $266,894
Amount of Taxes Collected (Oct 16 - May 18) $58,887,618

TOTO for the Theorist: A Better-Than-Fair Gamble

Fair Gambles

TOTO offers a better-than-fair gamble, regardless of what type of bet you place or the size of the prize pool. First, let’s define what a fair gamble is. A fair gamble is one with an expected payoff of zero. For example, a gamble with a 50% chance of winning $1 and a 50% of losing $1 gives an expected payoff of:

\[ Expected\ Payoff = (50\% \times $1) + (50\% \times -$1) = $0 \]

If one were to participate in a lot of fair gambles, he/she would make nothing in the long run. Then, by definition, if one were to participate in a lot of better-than-fair (positive expected payoff) gambles, he/she would earn money in the long run. We need two pieces of information to calculate the expected payoff of a gamble: the probability and payoff from each possible outcome of the gamble.

Payoffs

A simplistic approach to assess how fair a gamble TOTO is would be to calculate expected value using only the probabilities of winning and the prize for each PG. But, this approach is not ideal, because of a major complication: bet type. The bet type affects the expected value because certain bet types offer a player more chances at winning more money. For example, a System 12 bet gives a player 924 Ordinary bet combinations, while an Ordinary bet gives a player just 1 combination (of only six numbers). This means that (1) there is a higher chance of meeting the conditions of a higher PG from using a System 12 bet as compared to an Ordinary bet; and (2) when a System 12 bettor wins in a higher PG, he/she also wins in lower PGs because he/she has more combinations to match against the winning numbers.

Therefore, instead of using the table provided by SG Pools, we need to use the following prize table:

 

 

This table tells us how many of each PG prize we could win given a (1) matching combination and (2) bet type. We will use these for the simulation in the next section.

Probabilities

Now that we know the payoffs, we need the probabilities of achieving each matching combination (e.g. 6+1, 5, 3+1). The probabilities are not as simple as they look. A System 7 bet may have seven Ordinary bet combinations, but this does not translate to seven times the probability of an Ordinary bet. In fact, the effect on the probabilities of winning in the various PGs do not increase at the same rate. To see why, let’s compare a System 7 bet to an Ordinary bet with the same six primary numbers:

\[ Ordinary: \{1, 2, 3, 4, 5, 6\} \\ System\ 7: \{1, 2, 3, 4, 5, 6, 7\} \]

The combinations from the System 7 bet (let’s call them System 7 Combinations) under System 7 would differ from the Ordinary bet combinations by at most one number. Hence, the chances of obtaining a matching combination (e.g. 3 primary + 1 additional) for the various System 7 Combinations are correlated. If one of the System 7 Combinations has no matches, it’s highly likely that the six other System 7 Combinations would have no matches as well, and would have at most one match resulting from the one additional number under System 7.

To figure out the probabilities of achieving matching combinations using each bet type, I exhausted my admittedly poor knowledge of permutations and combinations, and resorted to another technique for estimating the probabilities: Monte Carlo simulation.

Monte Carlo simulation is used when you are able to regenerate the randomness of a certain phenomenon, and want to understand the distribution of possible outcomes. For this project, I simulated 50 million TOTO games for each bet type to determine the probability that each matching combination occurs. This involved the following:

  1. Choose numbers to bet. To increase the simulation speed, I chose 1-7 for System 7, 1 to 8 for System 8, and so on. It would have made no difference if I had chosen other combinations of numbers between one and 49 - all combinations of chosen numbers should result in almost-identical probabilities from the Monte Carlo simulations if we have sufficient iterations. Using the smallest consecutive numbers in the available set of 1-49 provided a benefit: I could set a condition to identify non-starting combinations. Specifically, if the smallest winning number against a System k bet was bigger than k-2, there could be at most two matches, and therefore, no prize. This condition enabled me to skip iterations that had no wins (90-97% of 50 million iterations per bet type).
  2. Set the winning combinations. For each simulation, I chose seven winning numbers at random - six were the primary numbers, one was the additional number.
  3. Match the chosen numbers to the winning combinations. For each simulation, the chosen numbers were checked for three or more matches with the six primary winning numbers. If there were fewer than three matches, the iteration was skipped (no prize). If there were at least three matches, the algorithm checked for the number of matches, and whether the chosen numbers matched the winning additional number.

The results of the simulations were as follows:

Prize Group Combination System 7 System 8 System 9 System 10 System 11 System 12
PG1 6 4.8e-05% 0.00019% 0.00053% 0.0013% 0.0028% 0.0056%
PG2 5+1 0.00028% 0.0011% 0.0033% 0.0082% 0.017% 0.034%
PG3 5 0.006% 0.015% 0.033% 0.062% 0.11% 0.18%
PG4 4+1 0.015% 0.038% 0.082% 0.15% 0.27% 0.44%
PG5 4 0.2% 0.37% 0.62% 0.96% 1.4% 1.9%
PG6 3+1 0.27% 0.5% 0.83% 1.3% 1.9% 2.6%
PG7 3 2.6% 3.8% 5.1% 6.6% 8.1% 9.7%

Net Expected Value vs. Prize Pool

Before we perform the calculations of net expected value for given prize pools, we need to state the assumption that there are no snowballing and cascading effects. Since not every PG has a winner in some games, prizes can snowball, thereby distorting the expected value of each gamble. With this assumption, we can treat each gamble as identical and pure.

I defined 2,300 gambles with total sales of $2 million to $25 million in steps of $50,000. I chose this range because TOTO games from October 2016 have never had a total sales value of less than $2.6 million, and few have surpassed a total sales value of $25 million. For each gamble, I did the following:

  1. Define the prizes for PG 1-4 based on the total sales. Define the standard individual prizes for PG 5-7 ($50, $25, $10).
  2. For each matching combination (e.g. 4 primary + 1 additional):
    1. For each bet type (Ordinary, System 7-12):
      1. Multiply the defined prize pools / individual prizes with the number of prizes in the prize table above.
      2. Add up the total prizes across PGs for that bet type.
      3. Multiply the probability of that combination with the sum of total prizes for each bet type.
  3. Sum up the numbers separately for each bet type in Step 2-a-iii to get the net expected value of that gamble, for different bet types. Save each net expected value separately.
  4. Go back to Step 2 and repeat until all gambles have been completed.

The results of the calculation are given in the graph below:

All bet types increase with the size of the prize pool. More importantly, all bet types had a positive net expected value. In other words, TOTO offers everyone a better-than-fair gamble regardless of the bet type they use. If players are offered better-than-fair gambles for all bet types, how does SG Pools make any money at all?

The answer is simple, and it comes from the definition of a better-than-fair gamble. The positive net expected value is only realised in the long run, after many, many TOTO games. Surely there aren’t enough players who participate in every TOTO game religiously over a prolonged period of time. A 100 games alone would take a full year (two TOTO draws a week) - that’s a heavy commitment, but still a small number of games relative to the 13,983,816 possibilities! Hence, short-term players do not play enough games to break even or profit through a massive win, and end up contributing to the prize pool for other lucky or determined players to win. I don’t blame them. It would take thousands of games and hundreds of thousands in costs to win in PGs 1-4. What could the outcomes be? We answer this question using simulation.

Simulation of Games

Suppose we play every TOTO game for 10 years: a total of 1,040 games. Let’s also assume that each game has sales of $3.7 million (the median in the dataset), and the winners don’t share any prizes. We simulate the above 1,000 times for each bet type to identify the possible outcomes. In each of the graphs below, the grey lines represent the net positions (balances) for the 1,000 simulations of 1,040 TOTO games each, and the red lines represent the median balances.

See the table below for summary statistics on outcomes, maximum balances, and minimum balances. The maximum and minimum balances refer to the highest and lowest observed net positions from the 1,000 simulations of 1,040 TOTO games. That is, you should be willing to dip as low as the given minimum balance and expect, at most, the given maximum balance during your 10-year TOTO marathon. For example, if I chose to play only System 10 for 10 years, I should be mentally prepared to lose as much as $186,740 over this period, I should accept that I could make at most $5.8 million, and I should expect to earn approximately $688,735 in total, which averages to $5.7k per month.

Bet Type Outcome Range Median Outcome Min. Balance Range Median Min. Balance Max. Balance Range Median Max. Balance
System 7 -$6,753 to $1,654,592 -$5,473 -$6,760 to -$7 -$5,495 $0 to $1,657,372 $0
System 8 -$25,522 to $2,300,158 -$20,942 -$25,550 to $0 -$21,130 $0 to $2,312,216 $72
System 9 -$75,076 to $4,041,134 $175,359 -$75,160 to $0 -$39,439 $0 to $4,055,888 $224,779
System 10 -$186,740 to $5,880,835 $688,735 -$186,950 to $0 -$47,188 $0 to $5,883,480 $749,635
System 11 -$391,433 to $8,327,032 $1,434,947 -$391,895 to $0 -$64,246 $0 to $8,416,884 $1,550,222
System 12 -$782,796 to $15,899,764 $3,398,654 -$783,720 to $0 -$78,624 $0 to $15,963,996 $3,525,697

Note that the median outcomes for System 7 and System 8 were negative. The reason for this is the low probability that these bet types would hit any big wins. It would take a lot more games to win in bigger PGs that would offset the observed losses after 10 years of gaming. In fact, the outcomes of -$5k and -$20k would easily be offset because they are small relative to the size of the rewards from PGs 1-4.

TOTO for the Realist: Forgo the Big Wins

As you would have seen from the probability table earlier in the sub-section Probabilities, it is incredibly difficult to meet the criteria for PG 4 and above. Hence, let’s examine the net expected value of a more realistic scenario: winning TOTO games in PGs 5-7, and treating PGs 1-4 as though they provided no prize money (i.e. forgoing any big prizes). This will give us a more realistic estimate of our net expected value because winning in PGs 5-7 is much more probable than winning in PGs 1-4.

Metric Ordinary System 7 System 8 System 9 System 10 System 11 System 12 iTOTO
Net Expected Value -$0.76 -$5.31 -$21.29 -$63.75 -$159.37 -$350.62 -$701.23 -$20.78
Return on Bet Cost -75.9% -75.9% -76% -75.9% -75.9% -75.9% -75.9% -63%

We find that there are no better-than-fair gambles for PGs 5-7. Based on these net expected values, all bet types except for iTOTO had a similar rate of return. Performing the same simulations above of 1,000 iterations of 1,040 games each, we see that TOTO is a losing gamble. We can expect to lose lots of money at a steady rate if we assume that we will never hit PG 4 and above.

Conclusion: Irrational is Rational

In this post, I’ve shown that TOTO can be a better-than-fair gamble, depending on which PGs you target and your time frame. Over the long run, TOTO is a winning bet. The rare payoffs are sufficiently large and the small payoffs are sufficiently common to give players a better-than-fair gamble, which is probably what draws irrational gamblers and rational players alike. What most will not realise is that it will take a very long time to realise this net positive expected value. Over the short run, TOTO is a losing bet. Most of us would not have seen enough games to land a large win.

Paradoxically, the rational approach to TOTO is stereotypically irrational: go all-or-nothing. It sounds ridiculous. The statistics tell us that we should dump all our money into the more expensive bet types (System 10-12), and continue to do that for a really long time. Either that, or don’t bother playing: don’t play small or over the short term.

The results of the “Theorist” simulation reflect the outcomes of playing over the long term. It is possible to land big wins, but it is risky. The bigger the bet you use, the higher the cost, but the sooner you could break even or make big bucks, assuming you are the sole winner of PGs 1-4. Long-term players may use the results table for the simulations to tailor their bets according to their risk appetite.

The results of the “Realist” simulation (forgoing PGs 1-4) tell us that the positive component of net expected value comes solely from the larger PGs, and that if we play the short game, we would surely lose. Hence, the statistics suggest that we learn from the irrational gambler: go big or go home.

I set out to discourage people from gambling in TOTO, but the data and math led me to become more intrigued than disillusioned with the game. This post could well lead players to playing bigger, more frequently, and over a longer period of time - in other words, to gamble more. What have I done?!

Gamble irrationally, folks.