Probability of winning a 6 from 49 lottery (Sportka)

How do you work out the probability of winning a lottery such as the Czech Sportka? Can you improve your chances? Let’s take a look. Sportka works like this: a bag holds 49 numbers from 1 to 49, and in the draw six random numbers are taken from it. Players try to guess these six numbers. What is the probability that we guess all six drawn numbers?

We start by working out how many possible combinations can be drawn. It is important to realise that no number can be drawn twice in one draw, so the numbers 1, 2, 3, 4, 5, 5 cannot come up. The second important point is that the order in which the numbers are drawn does not matter: if the numbers come out in the order 1, 2, 3, 4, 5, 6 one time and in the order 6, 5, 4, 3, 2, 1 another time, it is the same combination, and if we had picked those numbers we would win either way.

So we know that we cannot use k-permutations, where the order matters. We use combinations, where the order does not matter. We are asking how many different combinations there are when we choose six numbers from 49. We write this as a binomial coefficient:

$$(\genfrac{}{}{0ex}{}{49}{6})$$

The formula for working out this binomial coefficient looks like this:

$$(\genfrac{}{}{0ex}{}{n}{k})=\frac{n!}{(n-k)!\cdot k!}$$

where the exclamation mark stands for a factorial. Substituting the values, we get:

$$(\genfrac{}{}{0ex}{}{49}{6})=\frac{49!}{(49-6)!\cdot 6!}=\frac{49!}{43!\cdot 6!}$$

We can expand the numerator:

$$\frac{49!}{43!\cdot 6!}=\frac{49\cdot 48\cdot 47\cdot 46\cdot 45\cdot 44\cdot 43!}{43!\cdot 6!}$$

Now we can cancel 43!:

$$\frac{49\cdot 48\cdot 47\cdot 46\cdot 45\cdot 44\cdot 43!}{43!\cdot 6!}=\frac{49\cdot 48\cdot 47\cdot 46\cdot 45\cdot 44}{6!}$$

And now we just have to finish the calculation:

$$\frac{49\cdot 48\cdot 47\cdot 46\cdot 45\cdot 44}{6!}=\frac{10068347520}{720}$$

which equals

$$\frac{10068347520}{720}=13983816$$

So there are 13,983,816 different combinations (almost fourteen million) that you can bet on.

Probability of winning the jackpot

You win the jackpot when you guess all six drawn numbers. Since there are 13,983,816 different combinations, your chance of winning is 1 in 13,983,816, which gives you a probability of

$$\frac{1}{\mathrm{13,983,816}}\approx 0.0000000715$$

To express this as a percentage we multiply the probability by a hundred, that is, we move the decimal point two places to the right: 0.00000715%. That is not a very large probability. To put it in perspective, the Czech Republic has a little over ten million people. If every person there bet on Sportka with unique numbers, there would still be three million combinations that nobody had bet on, and there would be about a one in five chance that nobody would win anyway.

How can you improve your chances of winning?

You can’t. Unless you bribe the person who draws the numbers. Every set of six numbers has the same chance of being drawn. The numbers 1, 2, 3, 4, 5, 6 have the same chance of being drawn as 5, 14, 15, 38, 41, 48. The bag does not distinguish between a neat sequence such as 1, 2, 3, 4, 5, 6 and a random sequence such as 5, 14, 15, 38, 41, 48. It is only us humans who like a run of consecutive numbers better. In the same way, the bag does not remember the numbers drawn earlier. The fact that the numbers 5, 14, 15, 38, 41, 48 were drawn last week has no effect on the probability of the same six numbers being drawn again next week.

You have two options for helping yourself when you play. The first is not to pick the numbers that other people pick. That does not change your chances of winning, because it makes no difference whether one person or a hundred people picked a set of six numbers. But if you do win, your prize will be shared between fewer people. The best thing of all is to pick completely random numbers. The chance that somebody else picks the same random six numbers is low. But if, for example, you pick the same numbers that came up in the last draw, there is a good chance that other people used the same strategy and you would have to share any win.

The second option is not to play at all and to invest the money in something more sensible instead.