How to earn money from the lottery

After thoughts on How Not To Be Wrong by Jordan Ellenberg

Chapter 11 of the book illustrates a group of MIT students, who used math to beat their local lottery system, and found themselves a whole lot richer. This got me interested to look at one of Singapore’s most beloved pastimes - playing the lottery (thereby forming long queues - also Singapore’s favorite pastime). Through this analysis, I hope to answer the considerations of playing the lottery, and summarize into 4 takeaways that you can implement the next time you decide to buy a ticket.

(Long queues to purchase lottery tickets in Singapore)

1. If you intend to earn money, don’t play the lottery

The most popular lottery in Singapore is called “4D”. A 4-digit lottery system that selects 23 sets of winning combinations, made up of any 4-digit sequence (eg. 0123, 1122, 3334, 5555). It is clear that the probability of choosing a winning 4 digit number is indisputably low. There are 10,000 possible combinations, and only 23 winning combinations per draw. Thus, the probability of a single bet winning any prize is 0.23% (23/10,000). A 1 in 435 chance. That’s like having to throw 435 basketballs, with only 1 making the hoop! Even then, it does not yet consider the payout amount.

Taking into account the probability of each prize group and its corresponding payout, we calculate the expected value of placing a $1 bet in the lottery. In the following paragraphs, you will see that the expected value for each $1 bet is actually less than the bet itself. Technically, by simply not playing, your net present value is already higher than someone who bets.

Hence, if your intent is to have more money, the rational decision should be to avoid gambling.

2. If you have to play 4D, bet big

There are 2 bet types in 4D - big and small. The big bet type pays out when your bet appears in any of the 23 prize groups (top 3 prizes, 10 starter prizes and 10 consolation prizes). Comparatively, a small bet pays out a higher quantum for the top 3 prizes, but excludes any prize in the starter & consolation groups.

(Payout quantum of big & small bets)

If you had to play 4D, choose the big bet types instead. The expected value for a big bet type is higher than a small bet type. Even though the payout from winning a small bet type is higher, it only pays out if you are in the top 3 prizes, which excludes any rewards for the other 20 consolation & starter prizes. The higher prize quantum does not compensate for the lower probability of winning, resulting in a much lower expected value, almost $0.08 lower per $1 bet as compared to a big bet.

(Comparing the expected values of a big & small bet)

A common betting strategy among Singaporeans (ask any auntie or uncle on the street) is to place a $1 big and $1 small bet on your ticket, in the hopes of a higher upside when winning the top 3 prizes. Instead, your expected value would have been better off if both dollars were used for the big bet.

(Total expected value for a $2 bet)

A $0.079 difference in expected value sounds negligible, but it translates to a 6% difference! Hence, bet big if you have to play 4D.

3. If you seek the thrill of winning, play TOTO instead

Since the payout quantum and the number of prizes in 4D are fixed, your expected value will never exceed your bet amount, which should be the case for any well-designed lottery (except a few instances in the USA that was mentioned in the book).

Playing 4D to earn money is not a rational decision from both the probability & expected value perspective. However, if your intent is to seek the excitement and entertainment value of gambling, TOTO might be a better bet.

TOTO is a lottery system where users pick 6 distinct digits between 1 to 49 (eg. 5, 15, 28, 33, 34, 49). A quick calculation shows that the probability of winning a prize in TOTO is higher than in 4D.

(Total probability of winning any prize group in TOTO)

The total number of possible combinations can be derived by choosing 6 out of 49 digits without replacement, in other words [C(49,6)] = 13,983,816. Winning the group 1 prize entails choosing the single combination of numbers, out of a total of 13,983,816 ways. In calculating the odds for a group 2 prize, we need to select 5 of the 6 winning numbers [C(6,5)], and 1 additional number from the remaining 43 numbers [C(43,1)], with a 1 in 43 odds of getting the additional number right.

Applying these calculations, the total probability of winning any prize in TOTO is 1.86%, which is 8x higher than the 0.23% in 4D!

Does it mean you should play TOTO? This circles back to the calculation of expected values. While 4D assigns a fixed prize quantum to each prize group, meaning your expected value is independent of any external factors, TOTO works in a pari-mutuel format. This makes it difficult to calculate a definite expected value as we are unable to predict the actual number of winners, and the prize quantum changes due to the rollover of prize money.

To establish a benchmark for comparison, I calculated the expected values from recent draw results.

(Expected value calculations from the past 7 draws)

The expected value per person averages at $0.4126, which is lower than the expected value of 4D. Thus, there is a higher probability of winning a prize in TOTO, but the expected value would be lower.

4. If you do play TOTO, is there a way to maximize your chances?

Yes, in fact.

While the total prize pool for a regular draw totals to about $2.8 million, there are festive draws like the chinese new year special with prize pools as high as $30 million.

(Comparing the prize quantum & the number of winners between special and regular

draws)

As seen, a larger prize quantum entices more people to participate, which could mean having to split the prize pool with more people. However, there is a net positive effect.

(Expected value calculations from the past 6 chinese new year draws)

Over the past 6 chinese new year draws, the expected value averaged at $0.8181 (no previous data available as the earlier years had a different lottery format). Although it is still less than your $1 investment, the expected value for a TOTO special draw is higher than 4D ($0.659), or a regular TOTO draw ($0.4126).

The chinese new year draw in 2020 actually turned out to be a black swan event that made mathematical sense to bet! Surprisingly, the expected value in 2020 totaled at $1.1837, which is higher than the $1 bet amount! Nonetheless, we are unable to predict how many winners there will be in each group for upcoming draws.

The interesting takeaway is that the expected value from group 1 alone contributed to the majority of the total expected value. From the example in 2020, expected value in group 1 was twice as high ($0.8581), compared to the expected value of group 1 in previous years! The total expected value for 2020 exceeded its initial $1 bet because it was the only year with a single group 1 winner. On the other hand, the expected values between group 2 to 7 over the years have little fluctuations.

(Comparing the expected values between different prize groups)

Having only a single group 1 winner from a TOTO draw increases the expected value of participating. Hence, if you decide to play TOTO, find a number sequence that is very uncommon, which others are unlikely to pick. Avoid popular numbers like 7, 8, 13. Avoid memorable numbers like dates or vehicle number plates. Singaporeans have the superstition of betting on the number plates of vehicles involved in accidents, bridal cars and even funeral hearses. Avoid numbers that form nice sequences on betting slips e.g. the first number of every row.

Conclusion

To answer the question on how to earn money from the lottery:

1. If you intend to earn money, don’t play the lottery. The probability is too low.

2. If you have to play 4D, bet big. Your expected value is higher than a small bet.

3. If you seek the thrill of winning, play TOTO instead. The probability of winning a prize is 8x higher than in 4D.

4. If you play TOTO, only participate in special draws with large prize pools, and maximize your expected value by betting on uncommon numbers that others are unlikely to choose.

Ultimately, the lottery will not get you rich (at least in well-designed lotteries). Appreciate it for its merits of entertainment. Play a small amount, but take it as lost money. There is minimal use in placing multiple bets. Making 10 bets instead of 1 might increase your odds of winning by 10 folds, but the absolute percentage is still negligible.

Hope that this was useful, and got you one step closer to striking it rich!

References:

https://www.intmath.com/counting-probability/singapore-toto.php

https://medium.com/@GuanWei/singapores-toto-hong-pao-draw-ca3dd02b8450