TOTO: Winning over the long run — Go BIG or Go HOME

Introduction to TOTO

Lottery is part and parcel of many Singaporean families. Being “exposed to gambling” from young — us cousins in the family used to hang around our grandparents’ place every weekends and “tikam tikam” or guess numbers for their weekly bets. We picked individual numbers out of a jar for TOTO and 4D, sometimes we win most of the times we didn’t.

Most writings I encountered so far don’t go beyond the obvious — cost and odds of bet types or prize amounts. Playing the Devil’s Advocate, I thought it would be interesting to offer some of my own thoughts on top of the good work by predecessors on why higher bet types are prohibitive yet offer the best chance of positive returns over a long run.


This post in no way encourages gambling. Betting is still a game of luck and probabilities. TOTO is a gamble with huge variance. You could lose hundreds, or thousands of dollars if you make many huge and frequent bets. It is a sure-lose way of throwing away money as the probability of winning anything is 1.85%. Even going for exams without studying will get you more than average 2 points overall.

However, is this prejudice without looking much more in-depth into the gamble itself? The point of getting into a gamble anyhow, is to opt for the bet with the highest probabilities based on the maximum risk that you can take. Predecessors have unseated this prejudice as well with exploratory analysis and simulation.

TOTO is fairer than it seems: there is a way to maximise expected winnings with enough capital. And I don’t mean buying up all of the different combinations.

How to win TOTO

Simple enough, 6 (min) to 12 (max) numbers are picked between 1 — 49, through the various bet types:

  • QuickPick: The system generates 6 random numbers between 1 — 49.
  • Ordinary Bet: Select 6 numbers between 1 — 49.
  • System 7 to System 12: Select up to 12 numbers between 1 — 49. More numbers gives you more Ordinary Entries in one bet.
  • iTOTO: The system generates 12 random numbers between 1 — 49. It is a Quick Pick System 12 Entry divided into 28 units: Unit cost and prize amount is divided by 28.
  • System Roll: Select 5 numbers between 1 — 49. The remaining rolling number ‘R’ represents all remaining 44 numbers.
Bet typeNumber of
Ordinary entries
Quickpick /
System 77$7
System 828$28
System 984$84
System 10210$210
System 11462$462
System 12924$924
System Roll44$44

The prize amounts for the various prize groups are derived from Singapore Pools website.

How hard is it to win a TOTO prize?

Generally, betters are highly interested in efforts required to reap the rewards versus the costs of attaining it. A higher bet type will have more ‘ordinary’ chances, thus better odds and higher probability. It’s all co-related.

Odds and probability for Ordinary bet are mathematically derived. For System 7 to 12, they are derived from Monte Carlo simulation . 50 million TOTO games was simulated to figure out the probabilities of achieving matching combinations with each bet type.

A bettor has a 1-in-54 chance of winning any prize for an Ordinary bet and down to a 1-in-7 chance for System 12 bet over the long run.

Strategies for winning TOTO

In short, there is no sure-win strategy. Ignore all the high-low, odd-even, sum-of-numbers, frequency-of-numbers strategies out there. Each draw is independent from the next draw; any number drawn is independent from each other as well. Lucky numbers, popular numbers etc., they do not mean anything really. Trends arise after a long run and can be explained by statistics.

Taking data for the past 1254 games (Game 2341 to 3594), we have come up with some statistics to examine.

Odd/Even numbers combinations

Each number has an equal 1/49 chance of being drawn from the start. Similarly with 25 odd and 24 even numbers, there’s an almost equal chance of an odd/even number drawn with a very slight probability that odd number has a higher chance of being drawn.

Therefore we observe that having equal number of odd/even numbers drawn is the highest (37.6%). Having a odd/even combination of 2-4, 3-3 and 4-2 becomes statistically higher at 81%.

Without going into the high-low range or high-mid-low or even each number range grouping (1-9, 10-19, 20-29 etc), we find that it will statistically tend towards equal distribution.

Same number appear from the last draw (not same combination of numbers)

At the end of 1 draw (6 tries), the probability of a particular number being drawn is 0.1292 [1/49 x 1/48 x 1/47 x 1/46 x 1/45 x 1/44]. The probability of the particular number being drawn at the next draw as well is 0.01669 [0.1292 x 0.1292]. Since there are 49 numbers to draw, the probability of any number being drawn in the next draw is much higher.

We observe that having 0 or 1 number appearing from the last draw is roughly the same. Having 0 or 1 number appearing from last draw is statistically high at 85%. In reality, we know as well that it is more improbable for 2 or more same numbers to appear from the previous draw.

Similarly, one can plot the same numbers that had appeared within the past 2, 3 or 4 draws to notice the trend with enough data.

Is the set of (1,2,3,4,5 & 6) impossible to take place? Out of 13,983,816 combinations, all numbers or even combinations has equal chances of appearing; we just haven’t had enough actual games yet.

Here, we are more interested in examining results of probabilities stimulations.

Is TOTO a fair gamble?

A fair gamble is one where you have a 50-50 chance of winning $1 with an expected payoff of zero. If you were to gamble in countless numbers of fair gamble, you make nothing in the long run.

Expected Payoff = (50%×$1) + (50%×−$1) = $0

A better-than-fair gamble is one where you have a higher chance of winning $1 with an positive expected payoff (due to higher probability). Say a 55-45 chance of winning $1. It could also be one whereby you win more dollars than you lose in a rarer chance, leading to a positive expected payoff as well (due to higher payoff). Over the long run, you will earn money.

Expected Payoff (due to higher probability) = (55%×$1) + (45%×−$1) = $0.10
Expected Payoff (due to higher payoff) = (40%×$2) + (60%×−$1) = $0.20

TOTO is a better-than-fair gamble, regardless of the individuals’ bet type or prize pool size.

HUH? Now by this time, people will be scratching their heads. How can TOTO be a fair gamble? It is 1 out of 13,983,816 to win the jackpot and 1 out of 54 to win any group prizes with an ordinary bet. With the most expensive System 12 bet, the odds are brought down to 1 out of 17,857 for the jackpot and 1 out of 7 for any group prizes. No way is this 50-50 or even better, and everyone is bound to lose over the long run.

I thought it will be interesting to look at the overall picture and then delve deeper.

Is Singapore Pools making more money than it does giving out?

Just like a casino, we would expect that Singapore Pools is making much MORE money than it is giving out if TOTO was not a fair gamble. Is Singapore Pools making a huge profit all this while?

As a background, 54.5% of TOTO sales in each draw is for distribution of prizes in the Groups 1 — 4 which may be won for each draw. The rest of the TOTO sales is reserved for Groups 5 — 7.

Prize groupNumbers matchedPrize amount
16 nums38% of prize pool
(Minimum Guaranteed
of $1,000,000)
25 nums + 1 add8% of prize pool
35 nums5.5% of prize pool
44 nums + 1 add3% of prize pool
Total54.5% of prize pool

The prizes for the higher groups 1 — 4 are fixed as a percentage. The number of winners does not matter as the group prizes will be shared equally among the winners in the same prize group.

Prize groupNumbers matchedPrize amount
54 numsno. of winners X $50
63 nums + 1 addno. of winners X $25
73 numsno. of winners X $10
Total45.5% of prize pool

While the prizes for the lower groups 5 — 7 are fixed dollar amounts, the number of winners does matters as all winners get the same dollar amounts in the same prize group.

Singapore Pools makes money only if total payout for the lower groups 5 — 7 is less than 45.5% of total sales.

Interestingly, Singapore Pools has made almost nothing when looking at data from October 2016 onwards. The overall sales is negated by overall winnings from the betters. From this simple analysis, we can infer that TOTO is a fair gamble. By fair, we do not mean that the game is fair to you but as a single party (betters) to single party (Singapore Pools) comparison, both parties have an almost equal win in absolute dollars.

Bigger Prize Pool translates to Bigger net expected value

Prize Groups 1 — 4 are allocated 54.5% of the prize pool or total sales of TOTO. Generally, the larger the prize pool, the larger the winnings. All bet types encountered a positive net expected value but over the long run. In other words, a better-than-fair gamble regardless of the bet types.

The only issue is that people naturally “give up” after not winning for some time. We don’t see players participating in all TOTO games over a prolonged period of time.

The compulsive gambler too does not follow a religious system in their betting — they bet however big they want or “tikam tikam” over a few different bet types. They upped their bets after winning something, which sometimes turn their winnings into bigger losses instead.

Hence, I would agree that short-term players end up contributing to the prize pool for other lucky or determined players to win. These said players do not play enough to break even or profit through a massive win. It would take thousands of games and possibly thousands of dollars in costs to win the big Prize Groups 1 — 4. Even in 10 years of our lifetime, only 1040 games are played.

Simulation of 1040 games

To cut the long story short, predecessor had done 1,000 simulations of 1,040 TOTO games each for each bet type to identify possible outcomes.

The outcome could be summarised in the below table

The outcome range shows the maximum and minimum net balances from the simulations; you should be willing to be down as low as the minimum balance and expect at most, the maximum balance during a 10-year TOTO regime.

By betting on only System 9 for 10 years, I should be prepared both mind and soul to lose as much as $75,076 and make at most $4.0 million. As the median, I would expect to earn approximately $175,359 over ups and downs.

System 7 and System 8 turned out to have negative median outcomes instead, largely due to the low probability of hitting any big wins.

The above simulations were strict in a few criteria: no snowballing or cascading of prize monies, no sharing of prizes. In reality, these few factors do change the flavor in the expectations further.

A word of caution

Statistics however do not determine trends. Take a coin with Head/Tail sides. By taking 1 million simulations for example, we find that the outcome is roughly 50% for both Head/Tail. This does not determine however that one side cannot happen for a long time before the other side happens e.g. 20 Heads appear before 1 Tail.


The simulations for TOTO above is carried out for thousands of games. Over 10 years, only 1040 actual games are carried out — not quite a lot of games if you were to ask me. There is nothing against stopping you from losing tens of games before winning a small Prize Group, or even hundreds of games before winning a large Prize Group.

Yet, it does not change the fact that TOTO could be a better-than-fair gamble as the rare payoffs are sufficiently large (to compensate for the probability of winning them) and the smaller payoffs are sufficiently common. It also depends on the bet type you intend to stick to (for a long time) and your time frame.

Over the short run, TOTO is a losing bet for most of us who do not stick around for enough games to land a large win. Over the long run, TOTO becomes a winning bet, though it will take a very long time to realise a net positive expected value. Statistics is in your face to either gamble on the more expensive bet types (System 10-12) for a very long time or not bother at all.

It sounds ridiculous but rational at the same time. The only thing holding us back is our available capital. That is why you do not hear of many people even playing System 10 consistently over a long period of time.

Could we have it any worse?

Credits to the writer of TOTO: An Irrational, Rational Approach to Singapore’s Second Biggest Lottery for stimulating my thoughts.

Can Car Registration Number be a good investment?

In Singapore, like any other countries, every motor vehicle must have a vehicle registration number (VRN) to operate on the road. Each VRN is unique and like no other — it does not repeat and cannot be reused once discarded.

Most motorists use the VRN automatically assigned by LTA when buying a new vehicle. There is no charge for an assigned VRN. You could also bid (from LTA) for a new VRN or retain a VRN from an older vehicle. Your third option is to purchase and transfer a VRN from the open market.

History of Prefixes

A VRN generally consists of 3 parts: the prefix, the numbers and the suffix. Take ‘SJH1842U‘ for example. SJH is the prefix, 1842 form the numbers and U is the suffix derived from the checksum of the prefix and numbers.

Private car VRN began in the early 1900s, with a single prefix S for denoting Singapore. There are only 9999 numbers in each series. As the years went on, 2-letters prefix had to be introduced in 1913, E/EA introduced in 1972 before going back to S with 3-letter prefix in 1984.

At the time of writing, we are currently on SMU prefix.


Vintage plates on the road

If you are observant whilst on the road, you may have observed cars who have 2- or 1-letter number plates. These number plates are known as vintage plates, with the last of the series EZ beginning in 1984.

Car Number Plate “more expensive” than the Car

One of the most satisfying thing as a driver on the road is to notice nice numbers on another vehicle, even better if it is a vintage plate. It really hits-it-home if the number plate is “more expensive” than the car itself that it is affixed to.

The market value for a single-S car registration number is priced around $300,000.

Vintage vehicle registration number auctioned for $335,000

S32H at a public auction in Feb 2016 sold for $335,000. A similar S45X is offered for sale at $388,000 for sale on sgCarMart at that time. The auction or sales of vintage VRN is not new. Even now, older VRN are still highly priced and sought after.

Single-number plate bidding

The bidding for new VRN from LTA is good revenue for the government. Some $600,000 to $1 million in revenue is collected from 1,000 and 1,200 received bids for each series. The highest bid amount received so far has been $118,000 in 1995. All revenue is payable to the government’s Consolidated Fund.

Not all single-digits are equal — numbers 1, 3, 7, 8 and 9 usually attract the highest bids.

1 remains the most popular, with its significance is to be the “number one”. Equally nice-looking single-digits in a ‘normal’ series go for at least $10,000.

For the Chinese, they attribute numbers to fengshui.

2 (二, ÈR)

The number represents ‘double’ or ‘双’, which may also means a pair or doubles bring blessings.

3 (三, SĀN)

三 has similar sounding to 生, which means “life” or “to give birth” in Chinese. 生 has a positive connotation.

4 (四, SÌ)

四 has similar sounding to 死 (sǐ), which means “death” in Chinese.

5 (五, WǓ)

五 has similar sounding to 无 (wú), which means “not” or “without” in Chinese.

6 (六, LIÙ)

六 has similar sounding to 流 (liú), which means “flow” in Chinese. A popular Chinese idiom, 六六大顺 (liù liù dà shùn), denoting things will run smoothly.

7 (七, QĪ)

Lucky Seven has seen the number 7 as a lucky number from the western culture. 七 has similar sounding to 起 (qǐ), which means the “start” or “rise”, and also 气 (qì), which means “vital energy”.

8 (八, BĀ)

八 has similar sounding to 發 (fa), which means “wealth”, “fortune”, and “prosper” in Chinese. 8 is the luckiest number in Chinese culture. Multiples of eight are approved, 88 for example bears a resemblance to 囍 (shuāng xǐ), or “double happiness”.

9 (九, JIǓ)

九 has similar sounding to 久 (jiǔ), which means “long lasting” and “eternity” in Chinese.

Novelty plates

Sometimes, we find novelty plates on the road, and these gets tongues wagging due to their being checky.

Are vintage or nice numbers out of my league?

It really depends on how deep your wallets are. Yet, it need not break a hole in your pocket. Bidding for nice number plates is still a possibility with each series that come out from LTA.

For vintage plates, there are also those which are more affordable yet still don’t sound too bad.


Each VRN number is unique from one another. Once discarded, the same combination can never be reused. The supply of old or vintage VRN continues to fall over the years as people scrap their cars and give up their VRN.

The rich or those with expensive rides tend to go for a fancy VRN. The market demand is there for nice number combinations, single-digits and vintage VRN.

  • The older the prefix (except 3-letter prefix), the more valuable.
  • The shorter the prefix, the more valuable.
  • The lesser number of digits, the more valuable.
  • For single digit, 1 is the most valuable.
  • Next up on the popularity list for single digits in no particular order: ‘8’ means fa or prosperity in mandarin, ‘6’ sounds like luck when pronounced in dialect, ‘7’ for lucky seven, ‘9’ for longevity in mandarin, ‘3’ for life in mandarin.
  • 4 is the least valuable, since it means “die” in mandarin.
  • Next up on the least popular list in no particular order: ‘2’ and ‘5’.
  • Repeating numbers, especially those that are more auspicious sounding e.g. ’88’, ‘888’, ‘8888’, ’66’, ‘666’, ‘6666’, ’77’, ‘777’, ‘7777’, ’99’, ‘999’, ‘9999’.
  • For the rest of the other repeating numbers, though not as popular, they have a higher value than mixed numbers e.g. for repeating ones, twos, threes, fours, fives.
  • Novelty names or vanity plates e.g. SMA1180Y (SMALL BOY), SE1X, SMA5H (SMASH)

Generally if you have an old car running in the family, have a look whether it has a nice combination of the above. It is also not uncommon for number plates to be passed down from one car to another as the family retains the VRN.

Who knows if you have another S32H for the future?

Regular Savings Plan Update – September 2020

Monthly Allocation for 19 September 2020 as follows , with a $100 investment amount

CounterQuantityPriceInvestment Amount
OCBC Bank (O39)11$8.516$93.67

Average Price of existing Portfolio counters

CounterTotal SharesAvg Price
OCBC Bank (O39)460$10.2239
SIA (C6L)645$8.6553

Do I need to plan for Retirement till Age 99?

Retirement to me, is when as an individual, I am able to choose whether I “want” to work and not whether I “have” to work. Some people amassed enough to live by, based on their current lifestyle for the rest of their lives, but still remain in the workforce to keep themselves active. To possess this choice, you probably need to have:

  • Your own home which is fully paid up
  • No more debts
  • Adequate health and medical insurance coverage
  • Enough savings or continuous income flow to sustain your retirement lifestyle

The aged-old saying for early retirement planning still holds true: put time on your side… When time is your friend, the earlier you start the more you end off with. However, it is not uncommon to have doubts:

  • How much is enough really enough?
  • Will inflation erode my retirement funds faster than expected?
  • What if I outlive my retirement funds?

Do I really need to plan my retirement till Age 99?

Everyone wants to live a long life — while long is subjective, what they mean is probably to live life as fulfilling and satisfactory long as much as possible. Competing to be a centenarian is probably not on the minds of anyone, I believe.

So, does your retirement funds need to last you till age 99? The good news is, you don’t have to plan till so far. The bad news is, you are probably “gone” by age 83 plus or minus. The average life expectancy for Singaporeans is 83.6 years for year 2019, 82.6 in 2014 and 81.4 in 2009.

One one extreme, we have Singaporeans doing far too little for their future — their retirement plans won’t last beyond a few years due to ‘excessive’ lifestyle or pittance retirement pot. On the other extreme, there are few (but small numbers) who over-plan. I have to quantify, over-planning is never a bad thing — it’s a matter of whether you are sacrificing too much (when you are young) for too little additional (for when you are old).

If you retire at age 65, you would do well to ensure that your retirement pot lasts well for another 18 years. And so on for any other retirement age. Planning for another 34 years (till 99) sounds abit far-fetched now, seeing that it is not very probable.

Source: Department of Statistics (Singstat)

Do I have to plan for a longer retirement today as compared to the past?

Yes, yes, YES!

The infographics above has clearly shown that both male and female residents in Singapore are gradually showing a higher life expectancy. If we look at the population level, 16.4% of Singapore Residents are aged 65 and over in 2019, 9.1% in 2010 and 7.7% in 2000. The average age is inching higher year-by-year, from age 34.0 in 2000, to 37.4 in 2010 and 41.1 in 2019.

It is a undisputed fact that people are indeed living longer — whether it is due to better education, housing, healthcare provided today as compared to yesteryear.

Source: Department of Statistics (Singstat)

Females statistically live longer — the numbers double / almost double especially for the various 5-year age categories above 80.

How Singapore’s population demographics progressed over the last 20 years (2000 — 2019)?

I have included the progress of the population demographics over the past 20 years. The top (> age 65) is getting fatter, while the bottom (< age 15) thinner. It is a clear indication of an ageing society. It is good that people are living longer; it ain’t so good when it is accompanied with a dwindling birth rate.

Aging population

Source: Department of Statistics (Singstat)

The Resident Old-age Support Ratio shows how many young residents are present in relation to each older resident. In a span of almost 10 years, the ratio had dipped from 7.4 to 4.5. One direct impact will be on taxes. We expect most if not all older residents to fully retire from active work or at least have a chargeable income low enough not to warrant the paying of income taxes from age 65 onwards. This potentially places a higher tax burden on the future younger residents who have to pay more to maintain or even increased government expenditure on societal good.

We have not included the burden for schemes to take care of people who had fallen behind on their retirement funds for daily expenses or even for healthcare needs. It is on the government to take care of its citizens and ensure that no one gets left behind, yet we know that the source of funding still comes from its citizens.

What is my take for the future?

In comparison to my parents’ and grandparents’ generation, we have a better change of living longer, more healthier and better. Healthcare in Singapore is one of the top-notch in the world, coupled with early detection technology, the chances of discovering illnesses or recovering is also higher.

On a good gauge being age 35 today, there’s still a good 47 years or half-way through a life-long marathon.

Source: Department of Statistics (Singstat)

As an individual, I refuse to fall between the cracks during my retirement. In addition so as not to burden on society and the next generation, we really need to play our part as individuals. I do not want to create a future whereby my descendants are burdened due to the incapability of our generation to plan ahead. This is also an irony when people today are more well-educated and experienced as compared to the earlier generations.

Planning for retirement is not only (being responsible) for yourself, but also for your family and of course society. Life is short and as we go through the various stages in life, we find that there is only so much time to plan for it backwards and make it fulfilling as best as we can.