Lottery odds calculator

In a 6-from-49 lottery such as the Czech Sportka, six numbers are drawn from 49 and you pick six numbers of your own. You win the top prize by matching all six, and smaller prizes are for three, four or five correct numbers. The calculator works out the probability of matching exactly m numbers in any lottery where you pick as many numbers as are drawn.

How it works

The calculator uses this formula:

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

Here n is the number of numbers in the pool, k the number drawn (and picked) and m the number you want to match.

First we count how many different sets of six numbers can come out of the pool. The order in which the numbers come out does not matter, so this is a combination:

$$(\genfrac{}{}{0ex}{}{49}{6})=\mathrm{13,983,816}$$

There is only one way to match all six numbers, and that is to pick exactly the six numbers drawn. The chance of the top prize is therefore 1 in 13,983,816, which is about 0.00000715%.

For fewer matching numbers we also need to count the ways the unmatched numbers can be chosen. To match three numbers, we pick three of the six drawn numbers and the remaining three from the 43 that were not drawn:

$$P(3)=\frac{(\genfrac{}{}{0ex}{}{6}{3})\cdot (\genfrac{}{}{0ex}{}{43}{3})}{(\genfrac{}{}{0ex}{}{49}{6})}=\frac{20\cdot \mathrm{12,341}}{\mathrm{13,983,816}}=\frac{\mathrm{246,820}}{\mathrm{13,983,816}}\approx 0.0177$$

You will match three numbers roughly once in every 57 draws. For four numbers the chance is about 1 in 1,032, and for five it is 1 in 54,201. Sportka also has a bonus number, which decides the second prize, and the calculator does not count it. For a lottery with two pools, see the article on Eurojackpot.