How much you really lose when playing the lottery

Most states have some kind of lottery, and for a ticket the minimum price is usually $1. Let's take a generic lottery game where you pick four numbers, 0 through 9, for example 2278. If those are the winning numbers you collect and win $5000.

Now suppose you go play this game tonight and buy a single ticket. If you lose you just lost $1. If you win you collect $5000, minus the one dollar it cost you to play, but who cares about that right?

Ok so here is how you work it out, for any lottery game, first workout the expected value.

Expected Value = (What you win) *(Probability of Winning) + (What you lose)*(Probability of Losing)

For our game, if you win, you collect $5000. If you lose tonight, you only lose $1, so we'll count that as -1.

To figure out the probability of winning, note there is only 1 way to win and 10*10*10*10 = 10,000 ways to play, so the P(winning) = (number of ways to win)/(number of ways to play) = 1/10000


To figure out the probability of losing, note there are 9999 ways to LOSE this game, yikes!
This means P(losing) = 9999/10000


So finally,

Expected Value =   (5000)*1/10,000 +  (-1)*(9999/10000)
Expected Value = -.4999, so let's call that a $0.50 loss.

What does this mean? There is no way you can lose 50 cents if you play this game tonight, you either win or lose, so what in the world is this?!

Well there is something called the Law of Large Numbers that says that in the long run, how much you lose on average approaches 50 cents per game. Think about it, if you keep playing over and over you will eventually win, and you will also lose a lot, but in the long run, if you had an infinite amount of money and just played a lot every single day forever, your average loss per game gets closer and closer to 50 cents.

In other words, you are still guaranteed to be a loser, no matter what!

Of course if things weren't this way, the lottery would be out of business, there wouldn't be any casinos, nor would there be any gambling, because all of these institutions would fail in the long run.

I hope someone has found this post helpful!