Mar 06, 2019

In theory, it can only rely on luck. However, if the rules are not well designed, you can drill holes.

In February 2005, a lottery variety in the United States had a loophole that was discovered by MIT students. In the following seven years, the student repeatedly purchased this variety and earned a total of $3 million.

This article describes how he did it, and the mathematical principles in it. The material I rely on is mainly from a lecture by mathematics professor Jordan Ellenberg at Stanford University (Youtube).

First, the expected value

The most important mathematical concept of a lottery is called the "expected value", which is the average income that can be obtained after repeated repetition of the same behavior.

For example, if you need 2 yuan for each draw, assuming 200 draws, you can win once and the bonus is 300 yuan. Then, you spent 2,000 yuan, a total of 1,000 draws, 5 prizes, and a prize of 1,500 yuan.

In other words, the total income of 1000 draws is 1,500 yuan, and the average return per time is 1.5 yuan, which is the expected value. Its calculation formula is as follows.

Expected value = 300 * (1 / 200) + 0 * (199 / 200) = 1.5

The expected value is 1.5 yuan, but the cost of each draw is 2 yuan, so the net loss is 0.5 yuan.

As you can see at a glance, this thing is not cost-effective. The more you do, the less cost-effective it is. Occasionally buy a lottery ticket, it will be considered; if you continue to buy lottery tickets all day long, you will definitely lose a lot of money (the above example is 100 yuan per 200 losses).

In short, expectations are a key indicator of the lottery's return.

Second, the Massachusetts WinFall lottery

Massachusetts has a lottery variety called WinFall. Its rules are simple: 1 to 48, you guess 6 numbers, guess there is a prize.

Fourth prize (2 of 6 guesses): 2 yuan for bonus

Third prize (3 out of 6 guesses): 5 yuan bonus

Second prize (4 of 6 guesses): 150 yuan bonus

First prize (5 of 6 guesses): 4000 yuan for bonus

Grand Prize (6 out of 6 guesses): All remaining bonuses in the prize pool

In one issue, a total of 9.3 million lottery tickets were sold, including one special prize, one million yuan, 238 first prizes, 11,625 second prizes, 198,000 third prizes, and 1.368 million fourth prizes.

The calculation shows that the expected value of this lottery ticket is 0.798 yuan.

Expected value =

1 million * (1 / 9.3 million) +

4000 * ( 238 / 9.3 million) +

150 * (11625 / 9.3 million) +

5 * (198,000 / 9.3 million) +

2 * (136.8 million / 9.3 million)

= 0.798

The price of each lottery ticket is 2 yuan, but the average income is only 0.798 yuan, not even half of it. It can be seen that this lottery ticket is very uneconomical. So it’s not attractive, and the number of people buying this lottery is decreasing.

The state government is in a hurry because the government draws 20% from the lottery (0.4 yuan each). If sales are reduced, the government's revenue will also decrease. Therefore, in order to increase the appeal of such lottery tickets, the government decided to amend the lottery rules.

Third, the new rules

The new rule is that if there is no special prize in the current period (no one guesses 6 digits), the prize will be awarded to the winners of the first prize, the second prize and the third prize. The new prize amount of each prize is as follows.

First prize (6 in 5): 50,000 yuan

Second prize (6 in 4): 2385 yuan

Third prize (6 in 3): 60 yuan

Still use the previous winning rate to calculate the expected value.

Expected value =

50000 * ( 238 / 9.3 million) +

2385 * (11625 / 9.3 million) +

60 * (198,000 / 9.3 million) +

= 5.53

The price of each lottery ticket is still 2 yuan, but the expected value becomes 5.53 yuan. Buying such a lottery ticket becomes very cost-effective, and if you buy it in large quantities, you can get 2.5 times the income. The reason why the expected value is greater than the cost of the lottery is because the bonus pool also contains the remaining bonuses in the previous period.

A student at the Massachusetts Institute of Technology discovered this. He scraped 5,000 yuan to buy the lottery ticket, and the result was nearly 15,000 yuan!

Fourth, how to choose a number?

Now we know that the new rules of the lottery are profitable and can be purchased in large quantities. However, there is still a question, how should I choose the number to guarantee the income? In other words, in the 48 numbers, which 6 numbers should you choose to maximize the revenue?

After all, you can't buy all the lottery tickets because the lottery proceeds come from those who didn't win. You can only buy a portion of the lottery ticket and try to make the number you buy the most likely to win.

To simplify thinking, let us consider a simple situation. I guess three numbers in 1 to 7, and the bonuses are as follows.

Guess 3: Bonus 6 yuan

Guess 2: Bonus 2 yuan

Guess 1: no bonus

You can choose seven combinations at the same time (that is, buy seven lottery tickets). How do I choose a number?

Five, combination number formula

First, let us consider how many combinations of three numbers are in the seven numbers from 1 to 7. This is called mathematics in the combination number formula.

The combination number formula refers to the number of all combinations of n(n ≤ m) elements taken from m different elements, represented by the symbol c(m, n).

Its calculation formula is as follows.

c(m, n) = m! / n! * (m - n)!

In the above formula, the exclamation mark indicates a factorial, such as 4! is equal to 4 * 3 * 2 * 1.

According to the above definition, the combination of the three numbers in the seven numbers has a total of c (7, 3).

c(7, 3) = 7! / 3! * (7 - 3)! = 35

That is to say, there are 35 combinations of three numbers. We can list them all.

123 124 125 126 127

134 135 136 137

145 146 147

156 157

167

234 235 236 237

245 246 247

256 257

267

345 346 347

356 357

367

456 457

467

567

Above are all 35 possible combinations, you must choose 7 from them. Which seven should I choose?

Six, the best combination

The answer is the following seven combinations.

123 145 167 247 256 346 357

These seven lottery tickets will maximize your profits. Because, regardless of the final winning number, they can guarantee that you always get a 6 yuan bonus. If the winning number is 123, then you get the first prize of 6 yuan; if the winning number is 367, then the three lottery tickets 167, 346, 357 each guess two numbers, you have three small prizes, the total amount of bonus is also 6 yuan .

Looking closely at the seven lottery tickets, you will find that they are carefully chosen: each number appears exactly three times. This leads you to either a big prize or three small prizes.

Seven, geometric selection method

How were these seven lottery tickets selected?

There is a geometric way to do this very simply. The seven numbers are seven points, and they are connected by a straight line. There are only three points on each line, and each point appears on three lines. Drawing the shape above, you get seven lines (the inner circle is also a line). Then, record the number on each line and simply select seven lottery tickets.

The more rigorous proof is this: the seven numbers from 1 to 7 have a combination of 21 two numbers (C(7, 2)), which means that as long as you buy all 21 combinations, you can guarantee Three small prizes. Because there are three combinations of two numbers in the three winning numbers (for example, the winning number is 367, then 36, 37, 67 can be small and medium prizes). On the other hand, since each lottery ticket contains three numbers, that is, a combination of three types of two numbers, at least 7 lottery tickets can be used to cover all 21 combinations.

Eight, the actual strategy

Going back to the previous question, how should the Massachusetts lottery be bought?

As long as you guess 4 of the 6 numbers, you can win the second prize. As long as you buy all the combinations of the four numbers, you can ensure 15 second prizes (6 winning numbers have a total of 15 four combinations). C(6, 4)).

There are 194,580 combinations of four numbers in 48 numbers (C(48, 4)). Since a lottery ticket contains 15 combinations, it is enough to purchase at least 12,972 lottery tickets (194580 / 15 = 12972). A combination of all four numbers. If you are interested, you can write a program that calculates all the lottery tickets that contain the 194,580 combinations.

Buying 12,972 lottery tickets requires 25,944 yuan (12,972 * 2). According to the previous prize amount, the prize for the second prize is 2,385 yuan, then the 15 second prize is 35,775 yuan (2,385 * 15). Therefore, by investing 25,944 yuan, 35,775 yuan can be obtained without risk. Of course, the premise of this is that no one guesses the special prize in the current period, otherwise the bonus will be greatly diluted.

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