Using Mathematics to Win the Lottery

Lottery Basics

Many of you are probably familiar how lottery works.  A lottery is a game where a smaller group of numbers is chosen from a larger group. If you bet on the right combination, you win the jackpot prize, which is usually staggering.


Although there is a common concept about lottery, there are variations in different places or countries.  In this post, I will use ours as an example. In the Philippines, as of this writing, we have three types of lottery: 6/42, 6/45 and 6/49. Yes, you guessed it right, 6/42 means 6 numbers are randomly chosen from a set of numbers from 1 through 42. We use the 6/42 lottery in the following discussion.

In our country, the process of choosing numbers in a lottery is shown on television.  First, balls of the same weight numbered from 1 through 42 are placed in a machine-operated transparent container. Next, the balls are mixed using a blower located at the bottom of the container. These balls are light enough to be floated by the air from the blower. Finally to determine the winning combination, a button is pushed to eject 6 balls, one after the other. See video below.

Now, suppose we bought a ticket of the 6/42 lottery, what is our chance of winning? Is there a way that we can win for sure?

We will answer the first question later. For the second question, yes, there is a way and, in principle, it’s simple. Just bet on all the possible number combinations.

Okay, so you’re not a math major and you do not know what I am talking about. Well, let me explain it in plain language. Suppose a mini-lottery has 5 numbers, say 1 through 5, and two numbers are drawn to determine the winner. The pair, {1,2} is a possible bet as well as {4,3}.  It is clear that the order of the numbers chosen does not matter, so betting on {5,2} is the same as betting on {2,5}.

Finding all the possible pairs or combinations in the 2/5 mini-lottery is a bit easy.  As we can see from the table, if we choose two numbers from 5 choices, we have 10 possible combinations or pairs (see red text). The pairs in black texts are just the reverse of those in red.  We did not also include the pair having the same number, say (2,2), because in a lottery draw, drawn balls are not returned back into the container before drawing the next ball.

If we buy 10 tickets and bet on all the 10 pairs on the table, surely, we will win. We will use the same principle in winning the 6/42 lottery draw.

Systematic Betting

It is very hard to use the strategy above if we have six numbers taken from 42.  There is no easy way to tally our bets.  However, we can use a computer in finding all the possible number combinations. The method below maybe done:

  1. We can start with 1,2,3,4,5,6, then 1,2,3,4,5, 7, and then 1,2,3,4,5,6,8 all the way up to 1, 2, 3, 4, 5, 42.
  2. After that, we can increase the fifth digit by 1 and we start with 1, 2, 3, 4, 6, 7, and then 1, 2, 3, 4, 6, 8, and so on. We can do this all the way to 1, 2, 3, 4, 6, 42.
  3. When our sixth digit reached 42, we now increase the fifth digit by 1; that is, 1, 2, 3, 4, 7, 8; 1, 2, 3, 4, 7, 9 all the way up to 1, 2, 3, 4, 5, 7, 42.
  4. Every time we increase the fifth digit by 1, we get the sixth digit by adding the fifth digit by 1, and then keep adding 1 until the sixth digit reached 42. For instance, 1, 2, 3, 4, 8, 9; 1, 2, 3, 4, 8, 10 all the way up to 1, 2, 3, 4, 8, 42. In varying the fifth digit, our last number would be 1, 2, 3, 4, 41, 42.
  5. After the last combination is finished, we change the third digit: 1, 2, 4, 5, 6,  7 and so on…


Another mini-lottery has 5 balls and 3 balls to be drawn. List all the possible combinations. How many tickets should we buy for a sure win?

Winning the 6/42 lottery

In the 2/5 mini-lottery example above, we have two numbers chosen from a group of 5 numbers. This is equivalent to the combination of 5 numbers taken 2 at time.  We have learned from the Introduction to Combination post that 5 taken 2 at a time is equal to \frac{5!}{(5-2)!2!} = 10, and that can be verified from the table above. In the 6/42 lottery, 42 taken 6 is equal to \frac{42!}{(42-6!)6!} =5,245,786.  This means if we want a sure win, we must buy 5,245,786 tickets.

The Difficulties of Winning

There are, however, several difficulties in getting our fortune and becoming a millionaire, although it is definitely possible. Some of them are enumerated below.

  1. Ticket Price. One 6/42 lottery ticket is 10 pesos as of this writing (note that it may increase in the future). This means that today we will need 52,457,860 pesos (about 1,165,000 US dollars) to buy all the tickets. Now you really have to be quite rich to have that kind of money.
  2. Buying the Tickets. Assuming that we have the money, we also have a problem where to buy 5 million tickets. Suppose each lottery outlet has 5,000 tickets, we need to go to 1000 lottery outlets to buy all the tickets we need.
  3. Time Filling Out the Tickets. If you are going to fill out 5,245,786 tickets alone, then you will probably not be able to do it. Assuming that you can fill out 1 ticket in 1 minute, then 5,245,786 minutes is equal to 3642.9 days or nearly 10 years: that is, without doing anything. If you are going to spend 8 hours a day filling out the ticket, like a regular working employee, you will triple the time above. You will finish filling the tickets out in 30 years!
  4. Hiring People to Fill out the Tickets. Suppose we hire 100 people to solve problem 3, then, the 100 people will share the 3642.9 days x 24 hours/day = 87,429.6 hours. This means that each of the 100 person will work for about 874.3 hours, or about, 110 days in an 8-hour work day. So, if the salary of 1 person is Php450 a day, then, the money you need to pay them is 450/day x 110 days/person x 100 persons = Php4.9 million pesos. Well, it is not as expensive as the tickets, so we can afford it.
  5. Number Combinations. We should also have a way of organizing our number combinations. Note that in the five million plus tickets, we must fill out our numbers accurately and without duplicates. The mini-lottery above can be easily listed, but we will need a computer program to enumerate the 5,245,786 combinations. Well, we can definitely afford a programmer since we are ready to spend more than 50 million pesos.
  6. Prize money. Needless to say, if we want to spend 52 million pesos, the jackpot prize should be more than that.
  7. More than 1 Jackpot prize winners. Even if the prize money is more than 50 million, we still have one problem.  If another person wins, we are in big trouble.  If the jackpot prize, for instance, is 100 million pesos, and there are three winners, each of the winners will only bag 33 million pesos, and we are 19 million pesos short.

Now, you probably understand why many mathematicians do not bother to buy lottery tickets, despite the fact that they know how to win. The answer to our second question above is that the chance of winning is 1 in 5,245,786.

What does that mean?

That means that if you buy a single 6-42 ticket and bet randomly 5,245,786 times, it is likely that you will win only once (well technically, that is a bit wrong, but I cannot use a better analogy). And, if the chance is so small, why is there that there is always a winner in almost every draw?

The Chance of Winning

Millions of people bet on lottery in each draw, so it is likely, that one or more would win. If 3 million people bet different combinations in one draw, that is equivalent to one person betting 3 million times which results to about 60% probability of winning. That is the reason that most of the time, there are lottery winners.

It has been done

Despite the numerous constraints, in 1995(?), two computer scientists from United States flew to Australia just to bet on a lottery (I saw the documentary several years ago).  They used a computer to systematically bet on all combinations. One employee from a lottery outlet was surprised when they told him that they were going to buy 100,000 tickets. The problem was they didn’t have enough tickets, so they had not bet on all possible number combinations.

They won anyway.

There is no secret to win a lottery.

There are websites that lure and convince people that they have apps or formulas that increase the chance of winning a lottery. These websites are definitely FAKE. Any high school student with good understanding of basic probability will tell you the same thing.

The only secret to increase your chance of winning is to buy more tickets. The more tickets you buy, the higher your chance of winning. In the 6-42 lottery above, if you are going to buy 2, 622, 893 tickets, then you have a 50% chance of winning. That means that in two bets, you are likely to win one jackpot prize.



  1. I am not really sure the process of lottery betting done in other countries, but here, first we buy tickets, then fill up our numbers of choice, then return it to the lottery outlet for processing. They place the ticket in a machine, and the machine ejects the receipt containing our number combinations. The receipt will be our official ticket.
  2. Prices and exchange rate discussed in this article are based on the time this article was written. They may change  in the near future.
  3. Photos: Money by AMagill, The Thinker by Andrew Horne
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  1. Here in the UK its pretty much the same as your 6/49 although you can also win some prize money for picking some of the balls correctly. You can win 10 pounds if you get three numbers and then there are prizes split between the contestants for 4 balls and 5 balls which would increase your winnings if you bought all the tickets.

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  3. so sad to know my chance of winning.. this only means that i’m just wasting my time and money betting on lottery..

    • Yes I believe I have a way just play one set of number in quick pick on a slip and have many draw’s and wait and watch how they are picking the number’s but you got the many drawing’s and I have won this way for many year’s, i played on the mega lottery here in texas after the first two drawing’s i hit five dollar’s last nite and almost hit the first five, i miss two number’s by five and one being off to hit, One Million dollar’s also it’s the place you go play at too where do you feel the luckiest at playing your number’s.

  4. the lottery 6/55 has now reached 700 million pesos! about 20 million us dollars. what are the odds for this if we bet on all sensible combinations. we can skip consecutive number drawings like 1 2 3 4 5 6 coz surely its not gonna be drawn. each tiket costs 20 pesos so is it effective?

    • @lucky i am: There are 28, 989, 675 combinations; that is the number of tickets that you need to buy. Meaning, if you bet on all combinations at 20 pesos each, you will need P579,793,500 pesos. If you have 579 million pesos, you can bet on all combinations and will win about 120 million if you are the sole winner.

      1 2 3 4 5 6 surely will not be drawn?

      That is not correct. All combinations are equally likely to be drawn, so no combination should be skipped.

      • That is not true either No TIME in the history of powerball has 123456 been drawn.

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  11. man i like your idea it good but its still about luck
    In my country we play 5|90 lotto please how do u suggest i play that
    i will be very happy if u come up with something and in my country u can some how predict winning numbers with the previous result but i am not good at that .
    so please help

    • Hi Timi,

      There is no way that we can predict the numbers that will come up in a fair lottery. As discussed above, to increase your chance of winning, you have to bet on more combinations. It is probably more than a trillion trillion times harder to win in a 5/90 lotto than in a 5/42 lotto.

  12. try betting again mr. bautista, and blog if you already won atleast the 5 numbers out of 6 on 6/42/ or 6/49 or the 6/55. it may be believable.

  13. Maybe if you buy more combination numbers doesn’t mean you have a chance to win in the lottery. It is how luck would come in your way and not chance. More combination numbers are like googling numbers.

    • @JP Torres Javier

      Buying more combinations does not necessarily mean that you will win, it just increases your chance of winning.

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  15. i am Rexford from Ghana. i need your help to win lotto in Ghana. i need lotto numbers to try in Ghana.
    Here lotto is full of calculation and people find it difficult to win lotto here because of the calculation.
    waiting for u.