The Expected Value Of Gambling $1,000 On Various Games

The house always wins. We always hear this when we talk about gambling. But what about that cousin’s friend’s brother who made a killing? For every winner, there are multiple losers. On top of that, although they made money on their latest bet, they fail to take into account the previous times that they lost money. Do not believe anyone that claims that they can consistently win at gambling games with a house advantage. 

lottery 5050 meme

Source: r/memes

Contrary to “popular” belief, winning is not 50:50 when gambling. The house has to make money and they do so via a house advantage. In this article, we will look at what are the expected values of various gambling activities.

Disclaimer on math: I tried my best regarding the calculations of the expected value of the various gambling games. Do let me know if there are any mistakes.

How Do Singapore Pools & Casinos Make Money?

The Unbeatables I 双天至尊 I - Episode 1

Source: MakeAGIF

Legitimate outlets do not need to cheat to make your money. They have a special sauce called the house advantage. The house advantage will be used to pay for various costs of operations like taxes, wages, rent and free rooms and the balance will be their profits.

House Advantage

Also known as the house edge, margin or commission among others, gambling outlets include a house advantage into their odds to make sure that they win over a long period of time and multiple bets placed. In the long term, this house advantage gives the gambling outlet a positive expected value and the gambler a negative expected value.

Do take note that this house advantage does not include the direct and indirect costs of providing the gambling service.

Direct costs include printing of betting slips, GST, lottery tax and casino tax while indirect costs include operating costs like rental and wages.

Expected Value

Expected value = [Odds x Outcome 1] + [Odds x Outcome 2] + … + [Odds x Outcome N]

An expected value is the long-term weighted average of each outcome based on its probability. In the long run aka after multiple bets, the expected value is the outcome you should expect. For example, if the expected value is 0.5 (coin flip), in the long run, you should end up with 50% heads and 50% tails. In the short term, you might experience variability and get 20 heads in a row. However, the more games you play, the variability will drop and eventually you will end up with 50% heads and 50% tails.

Fair Odds

To demonstrate, we will use a coin flip as an example.

Fair Odds – Betting S$1 On Heads

Outcome Probability Payout S$ Pb x Payout
Heads 1 in 2 2 1
Expected Value 1
House Advantage 0%

If this gamble has fair odds without a house advantage, it will be as above. If 50 people bet on heads and 50 people bet on tails (total S$100), the total payout is also S$100. The organizer of the coin flip (a.k.a house) doesn’t make any money and everything is paid out to the gamblers.

Unfair Odds

Unfair Odds – Betting S$1 On Heads

Outcome Probability Payout S$ Pb x Payout
Heads 1 in 2 1.95 0.975
Expected Value 0.975
House Advantage 2.5%

In comparison, the above table showcases odds with a house advantage. It is “unfair” in the way that it does not pay out according to the odds, resulting in a house advantage. “Unfair” in this scenario does not mean cheating. If you gamble, you should understand the house advantage before placing your bets.

Instead of getting a payout of S$2 (fair odds), we will only get back S$1.95. If 50 people bet on heads and 50 people bet on tails (total S$100), the payout is now S$97.50. For every S$100 bet, the house will make an average of S$2.50, a 2.5% house advantage. In the long term, if they receive bets totalling S$1 million, they will make about S$25,000 with some variation. On the other side of the table, gamblers as a whole will be S$25,000 worse off.

This is not to say that there are no players that have a positive expected value. However, the longer one plays, the higher the chances of reversion to the mean, leading to a negative expected value. Additionally, for every outlier (winner), there will be many (losers) who fall within the rest of the bell curve. If they play long enough, eventually, the only winner will be the gambling outlet.

Risk Management

Besides having the house advantage, gambling outlets also have other risk management strategies to prevent loss spikes due to freak results.

Closing Off Bets

If too many people buy a certain 4D number (e.g. Joseph Schooling’s timing) and it comes out as the top prize, Singapore pools might be in big trouble. This “hot” number was sold out within an hour. For every draw, there is a sales limit for each number to limit any potential losses. By closing off bets, it limits the maximum payout for each number, limiting losses and preventing freak outcomes.

Adjusting Odds

Sports betting adjusts its odds as time goes by. They can adjust the odds live if there is new information (e.g. player injury) or when certain bets are under or oversubscribed. Similar to closing off bets, they are able to control their losses.

Betting Cap

Technically, if you have infinite funds, you can eventually come out on top when you keep increasing your bets and leave when you win. The problem is that increasing your bets can go south very quickly. If you double your bets starting from S$1, it can quickly hit S$1,048,576 after 21 bets. Additionally, the table minimum for casino tables might start from S$20, making it hit the S$1,000,000 mark in 17 bets. Although it is unlikely that you will lose multiple times in a row, however, it is not impossible. And if you have S$1,000,000 to gamble, you probably wouldn’t start betting at S$1 or S$20. Even if you are hyperrational, and can start at low bets despite having a huge base, the casino will have a table limit to prevent you from betting over that amount.

Popular Gambling Activities

Here is a list of popular gambling activities and their expected values after betting S$1,000.

4D – S$341 to S$420 loss

To play 4D, we bet on 4 numbers ranging from 0000 to 9999. There are 23 prizes in each draw with different payouts for the different prizes as seen below. One can also bet on “Big” paying out on all prizes or “Small” paying out a bigger amount only on the top 3 prizes. The minimum bet is S$1.

Betting S$1 On Big

Prize Probability Payout S$ Pb x Payout
1st Prize 1 in 10k 2,000 0.2
2nd Prize 1 in 10k 1,000 0.1
3rd Prize 1 in 10k 490 0.049
10 Starter 10 in 10k 250 0.25
10 Consolation 10 in 10k 60 0.06
Expected Value 0.659
House Advantage 34.1%

Betting S$1 On Small

1st Prize 1 in 10k 3,000 0.3
2nd Prize 1 in 10k 2,000 0.2
3rd Prize 1 in 10k 800 0.08
Expected Value 0.58
House Advantage 42%

If you bet S$1,000 on “Big” cumulatively, the expected value is about S$659 (S$341/34.1% loss)

If you bet S$1,000 on “Small” cumulatively, the expected value is about S$580 (S$420/42% loss)

TOTO – S$349 Loss

To play TOTO, the basic bet is to buy 6 numbers out of 49 numbers. Each set of numbers costs S$1. There are also other bets that allow you to buy multiple numbers like system 7 – 12 which costs more. There is no “bulk discount” when you purchase more numbers in each bet. For example, buying System 7 for S$7 is the same as buying the 7 sets of 6 numbers.

For today’s calculations, we will be using the 2023 Hong Bao Draw. The expected value will be different depending on the prize pool and the number of people sharing the top 4 prizes.

Buying a S$1 TOTO Ticket – based on only 1 winner per group

Prize Probability Payout S$ Pb x Payout
Group 1 1 in 13.99m 12.8m 0.92
Group 2 1 in 2.33m 2.7m 1.16
Group 3 1 in 55,491 1.85m 33.41
Group 4 1 in 22,197 1.01m 45.61
Group 5 1 in 1,083 50 0.046
Group 6 1 in 812 25 0.031
Group 7 1 in 61 10 0.164
Expected Value 81.34

The expected value is overly positive at above 80, however, the top 4 prizes are shared between multiple winners. There might be multiple winners that chose the same 6 numbers in their betting slip. s, the expected value will be different.

After Splitting

Buying a S$1 TOTO Ticket – split between winners per group

Prize Probability Payout S$ Pb x Payout
Group 1 1 in 13.99m 12.8m / 3 0.305
Group 2 1 in 2.33m 2.7m / 22 0.053
Group 3 1 in 55,491 1.85m / 991 0.034
Group 4 1 in 22,197 1m / 2,428 0.019
Group 5 1 in 1,083 50 0.046
Group 6 1 in 812 25 0.031
Group 7 1 in 61 10 0.164
Expected Value 0.651
House Advantage 34.9%

After accounting for the sharing of prizes, the expected value drops sharply to 0.651.

If you spend S$1,000 on TOTO cumulatively, the expected value is about S$651 (S$349/34.9% loss)

Singapore Sweep – S$497 loss

To play the Singapore Sweep, you will need to choose 7 numbers from 1,000,000 to 4,499,999 (3.5m numbers). Unlike 4D and TOTO, each number is unique and cannot be bought by multiple people. Each set of numbers costs S$3.

Buying a S$3 Singapore Sweep Ticket

Prize Probability Payout S$ Pb x Payout
1st Prize 1 in 3.5m 2.3m 0.657
2nd Prize 1 in 3.5m 500k 0.143
3rd Prize 1 in 3.5m 250k 0.071
10 Jackpot
10 in 3.5m 10,000 0.029
10 Lucky
10 in 3.5m 5,000 0.014
30 Gift 30 in 3.5m 3,000 0.026
30 Consolation 30 in 3.5m 2,000 0.017
50 Participation 30 in 3.5m 1,000 0.009
315k Delight 315k in 3.5m 6 0.54
Expected Value 1.51
House Advantage 49.7%

Although the expected value is S$1.51, however, each ticket costs S$3.

If you spend S$1,000 on Singapore Sweep cumulatively, the expected value is about S$497 (S$503/50.3% loss)

Sports Betting – S$90 – S$460 loss

For sports aka soccer betting, there are various types of bets on can place.

Betting S$1 on Chelsea Vs Fulham on 4 Feb 2023 (1X2)

Result Payout Implied Odds
Chelsea Win 1.58 63.3%
Draw 3.6 27.8%
Fulham Win 4.7 21.3%
Total Probability 112%
House Advantage 12%

I checked on the basic bet (home/away/draw) for multiple matches and the typical house advantage is about 12%.

Betting S$1 on various bets – Chelsea Vs Fulham on 4 Feb 2023

Bet Type House Edge
1X2 12%
½ Goal 10%
Total Goals Over/Under 2.5 9%
Will Both Teams Score 10%
Halftime-Fulltime 25%
Pick The Score 46%
Total Goals 30%
Team To Score 1st Goal 10%
Total Goals Odd/Even 10%
Halftime 1X2 12%
Halftime PTS 31%
HT Total Goals Over/Under 1.5 9%
HT Total Goals Odd/Even 10%
HT Total Goals 15%
HT Team To Score 1st Goal 10%
HT Will Both Teams Score 9%

As for the rest, the odds will vary slightly between matches as above.

Depending on the type of bet placed, for every S$1,000 placed, the expected value is about S$540 (S$460/46% loss) – S$910 (S$90/9% loss).

Horse Betting – Up to S$250 loss

Horse betting is a bit different from the other lotteries where the odds are directly related to the amount of bets placed, similar to the top 4 prizes for TOTO. Depending on the type of bet, Singapore Pools take a commission ranging from 0% – 25% out of the total pot first and then pays out the balance to the winners.

Depending on the type of bet placed, for every S$1,000 placed, the expected value is about S$750 (S$250/25% loss) – S$1,000 (S$1,000/0% loss).

Casino Games – S$5 – S$333 loss

Here are some of the most popular games at the casino with the respective house advantage.

Casino Game House Edge
Roulette 5.26%
Baccarat 1.01% – 15.75%
Blackjack 0.5%
Slots 2% – 15%
Sic Bo (Big Small) 2.78% – 33.33%


For table games, the odds are overly complicated as they can depend on strategy and bet size. If you are interested in the odds for the various games, you can google them for yourself. I just took the edge from As for slots, it is set by the casino as there are no fixed odds. The point I am trying to make is that there is a house edge for the casino.

You might be able to vary your bet size and betting frequency to lower the house advantage using various strategies like counting cards, however, the casino can catch on to you quickly and switch decks or even ban you if you are too good at the game.

Depending on the game and how good you are at the game, for every S$1,000 placed, the expected value is about S$750 (S$250/25% loss) – S$1,000 (S$1,000/0% loss).

Social Gambling – Depends

With social gambling, the expected value can vary. Games like mahjong and texas hold’em are skilled-based where in the long run, the better player will win. In this case, the better player will have a positive expected value while the weaker player will have a negative expected value.

However, in games like blackjack, it is similar to the casino where the dealer has the advantage. In this case, taking turns to be the dealer will even out the odds.

In any case, only gamble with people you can trust. There are cheaters out there who can collude or even use special contact lenses to view markings on cards.

Why Do Some Gamblers Lose All Their Money?

I was thinking that since the house advantage isn’t 100%, why do some gamblers lose all their money? Shouldn’t they end up with the expected value? Here are some of the reasons that I have thought of.

Bad Risk Management

First, the gambler’s bet size is too big compared to their gambling funds. It can either be due to the minimum bet amount, greed or bad betting strategy. The higher number of bets, the higher the probability of reaching the expected value. If the number of bets is low, the expected value variability is high. When losing, bet sizes might get bigger and bigger to make back their losses, resulting in a lower bet capacity. They go bust before the results can revert to the mean expected value.

Next, they also don’t stop when winning. By chance, one might get ahead of the odds and end up with a temporary positive expected value. However, the gambler might feel that they are on a streak or that they have not gambled enough, so they will continue gambling. A new gambler might also think that they will be able to replicate their results when they win. They might even increase their bets to increase their projected earnings. By continuing to gamble, the expected value will eventually revert to negative due to the house advantage.

Lastly, the gambling funds eat into the gambler’s living expenses. Gambling expenses are discretionary while living expenses are typically fixed. Basic living expenses like food, utilities and transport cannot be cut beyond the minimum amount for survival. If gambling expenses eat into living expenses, the gambler might go into debt or even crime to survive instead of cutting down their gambling. This can lead to a death spiral where they spend their money on paying back debt, lose their jobs or even gamble more to try to break even.

Spend Winnings On Discretionary Items

When a gambler makes money, they might spend the money on luxury goods or treat their friends and families to a meal. On top of that, even distant relatives will appear to borrow money from them. The winnings that are spent will be significantly reduced or permanently removed from one’s net assets. By spending the money on discretionary items, any temporary positive expected value is now gone and there are now lesser funds. With lesser funds, there are fewer bets one can make, increasing the chances of going bust before reaching the expected value. So regardless of whether the gambler wins or loses, money is spent.

Need Help With Gambling Addiction?

If you or your loved ones need help with gambling addiction, here are 2 national organizations that you can approach.

National Addictions Management Service


Hotline: 6732 6837

Webchat available

National Council On Problem Gambling


Hotline: 1800-6-668-668

Webchat and counselling services are available

Self, family and third party exclusion applications are also available


The long-term expected value of gambling is negative for all gambling games with a house advantage. The house advantage is what the gambling outlet makes and is used to pay for its expenses, any balance will then be its profits. Despite the expected value not being zero, due to bad decision-making, some gamblers can still lose all their money.

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