Probability of winning the Eurojackpot

The Eurojackpot is a number lottery similar to the Czech Sportka, which several European countries take part in. The minimum guaranteed jackpot is 10 million euros and the maximum possible jackpot is 120 million euros. The rules are similar to those of a 6 from 49 lottery.

Rules

The aim of the lottery is to guess five drawn numbers out of 50 possible numbers and, on top of that, to guess two further bonus numbers out of 12. If you guess all seven drawn numbers, you win the jackpot, the Eurojackpot. If you guess fewer numbers, you can win smaller prizes in the lower prize tiers. Interestingly, in 2015 somebody in the Czech Republic won the main Eurojackpot, a record win of 2.466 billion Czech koruna (about 90 million euros).

Probability of winning the Eurojackpot

But what are your chances of winning? We start by working out how many possible combinations can be drawn. We know that no number can be drawn twice and that the order in which the numbers are drawn does not matter. So we cannot use k-permutations for the calculation, and we have to use combinations. We are asking how many different sets of five can be drawn from the 50 possible numbers. We write this as a binomial coefficient:

$$(\genfrac{}{}{0ex}{}{50}{5})$$

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:

$$(\genfrac{}{}{0ex}{}{50}{5})=\frac{50!}{(50-5)!\cdot 5!}=\frac{50!}{45!\cdot 5!}$$

Using the properties of the factorial, we expand the numerator:

$$\frac{50!}{45!\cdot 5!}=\frac{50\cdot 49\cdot 48\cdot 47\cdot 46\cdot 45!}{45!\cdot 5!}$$

We can see that both the numerator and the denominator contain 45!. So we can cancel the whole fraction by 45!:

$$\frac{50\cdot 49\cdot 48\cdot 47\cdot 46\cdot 45!}{45!\cdot 5!}=\frac{50\cdot 49\cdot 48\cdot 47\cdot 46}{5!}$$

We expand 5!:

$$\frac{50\cdot 49\cdot 48\cdot 47\cdot 46\cdot 45!}{45!\cdot 5!}=\frac{50\cdot 49\cdot 48\cdot 47\cdot 46}{5\cdot 4\cdot 3\cdot 2}$$

We multiply everything out:

$$\frac{50\cdot 49\cdot 48\cdot 47\cdot 46}{5\cdot 4\cdot 3\cdot 2}=\frac{254251200}{120}$$

And now we just divide:

$$\frac{254251200}{120}=2118760$$

So there are 2,118,760 different sets of five that can be drawn from the fifty numbers. The probability of guessing exactly the one set of five that is drawn, let’s call it P₁, is the reciprocal, that is

$$P_1=\frac{1}{2118760}$$

Now we need to work out how many different pairs can be drawn from 12 numbers, so that we can find the probability of hitting the bonus numbers as well. We can again write this as a binomial coefficient:

$$(\genfrac{}{}{0ex}{}{12}{2})$$

This time we skip the working and just find that

$$(\genfrac{}{}{0ex}{}{12}{2})=66$$

So there are 66 different pairs. The probability that we hit the correct pair, let’s call it P₂, is

$$P_2=\frac{1}{66}$$

We get the overall probability P of winning the Eurojackpot by multiplying these two probabilities:

$$P=P_1\cdot P_2=\frac{1}{2118760}\cdot \frac{1}{66}=\frac{1}{139838160}$$

The chance of winning the Eurojackpot is 1 in 139,838,160. To put it in perspective, the Czech Republic has a little over ten million people. To make certain that somebody from the Czech Republic wins, every Czech resident would have to buy 13 unique tickets. After the draw it would then be certain that one of the Czech tickets was the winning one.