The Math Behind Why The House Always Wins

As a part of the course I took this past semester on teaching math in secondary schools, I (with a small group of my classmates) taught the class a unit on data analysis and probability. I also had to write a few short papers about certain things we covered in our lesson. What follows will be a merged summary of two of my papers.

Before I begin I’ll provide some vocabulary for those of you that never learned these terms or learned them and have since forgotten them.

Expected Gain: How much money we should expect to gain if we continue to gamble on one game for a long time.

Fair Game: A game in which the expected value is equal to zero. In other words if we played a fair game for a long time we should break even.

Unfair Game: A game in which the expected value is positive (we will profit from the game) or negative (we will lose money by playing the game).

Now, let’s consider a game where you win $4 if you roll a fair six sided die and it lands on “1” but lose $1 if it lands on any other number. Would you be tempted to play this game because winning $4 is more than loosing $1? The way we calculate the expected gain of this game is to multiply the chance of winning (1/6) in one play by the amount we would win ($4) in one play and add that to the the chance of loosing (5/6) in one play multiplied by how much we gain for loosing (-$1). The equation looks like this:

(1/6)($4) + (5/6)(-$1) = -$0.16666….

Our expected gain is equal to about negative 17 cents. Basically it means that if we continue to play this game for a long time, we should expect to lose 17 cents on average. If we won $5 for winning instead of $4 the expected gain would be 0 and this game would be fair.

Now you might be thinking that my example was stupid because no casinos have a lame game like the one I mentioned. If you are, I have two points to make (1) idiots will bet on anything, and (2) they might not have this game but they have others like it. Consider the game of roulette.

There are plenty of casino games I could discuss, but roulette is the simpliest to understand. There are many ways you can bet on the roulette wheel, evens and odds, red and black, a third, a single number, etc. I’ll discuss the expected gain for when we place our bet on evens or odds. The payout is 1:1 (if we bet $1, we win $1 and if we loose we loose the $1 we bet).  Do you think our chance of winning is 50:50?

If you think that half of the wheel is even and half of the wheel is odd you’re wrong because you’ve failed to noticed the two green “0” slots. If you think that more of the numbers are even because in mathematics zero is, in fact, an even number you’re also wrong because it’s not that way in the casinos. Did you notice that our chances of winning are not 50:50, yet our payout is 1:1? Our chances of winning are 17:36. The expected gain in this case is:

(17/36)(1) + (19/36)(-1) = (-323/630) =  -0.05555…

This means that on average we can expect to lose about 5.6% of what we’re betting.

The reason that the house always wins is because *every* game they have is unfair in their favor. The only exception is Blackjack (21), and it’s only an exception if you can count cards. They either rig the odds or payouts of every game so that they have some sort of advantage over you. Even if we win big, it doesn’t hurt the casino because our win is made up for by all of the other player’s losses. So there you have it, the math behind why the house always wins.


2 Responses so far »

  1. 1

    Scott Pool said,

    This is why we need to learn to count cards ;D

  2. 2

    Very interested if this was applied to any of your classes!

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