A simple guide to playing the lottery

Playing the lottery is a simple investment: only if you partake you can win.
Many lotteries I know offer tickets with up to ten fields you can fill out.
Does that make economic or statistical sense? The answer is a clear "No".

Let me explain why that is so:

If you do not play at least one field, you cannot participate and you cannot win.

Playing at least one field enables you to possibly win. To keep things simple, I will only look at the jackpot, the highest win.
Your chances to rake it in big time are as follows with two popular lotteries:

Lotto 6aus49 1 : 139.000.000

EuroJackpot  1 : 95.344.200

Let us look at Eurojackpot with a hyypothetical jackpot of 100 Mio. Euro. It actually never has been that high so far.

You pay 2,5 € for a single lottery-field played to get a 1:96 Mio. chance to win 100 Mio. €. The expected value of your investment thus is 100 Mio. € / 96 Mio. which equals some 1€.

Let us see what happens when you play two fields: You invest 5€ and your expected value is 100 Mio. / 96 Mio * 2 which equals some 2€.

You can continue to look at that until you probably arrive at investing 250€ and looking at an expected value of 100€.

Keep in mind that your typical jackpot with Eurojackpot, however, is much lower than a 100 Mio. Euro, so the gap between your investment and the expected value is much larger than in the example shown above.

So, should you play the lottery? The answer is simple: if you do not play at all, you cannot win.
How much should you invest? Obviously as little as possible. One field played results in
a chance of 1:96 Mio. Two fields played net a chance of 2:96 Mio. and for that extra
1:96 Mio. you pay again 2,5€. That does not seem to make sense!

Now, if for some reason that jackpot swells up to some 250 Mio. € the story reads a tad bit differently. You could actually buy 9.534.420 lottery tickets and fill them out with every possible combination. If one of yours is right and nobody else has chosen the same numbers as you did, you would win 250 Mio. by only having invested some 238 Mio. in lottery tickets. That is a pretty nice cut of 12 Mio., but until that occasion arises, stick to playing one field, if played at all.
(idea for last para taken from: https://mostlymath.wordpress.com/2010/07/03/winning-the-lottery/ )