Wednesday, March 5, 2008

Wednesday Math, Vol. 17: Expected Value

Any business that works with risk, from casinos to insurance companies, keep a profit margin going because of a mathematical idea called Expected Value, or EV for short. Sometimes, there are a lot of possible outcomes for an event with a lot of different payoffs, but for this discussion, let's assume we are playing a simple game like roulette, where the two outcomes are winning and losing, and the payoffs are defined in advance. All we need to know is the probability of winning p, the Profit made when we win and the amount of money we Risk when we lose.

For example, if we bet that any single number comes up, let's say 8, the probability of winning p = 1/38, if we assume that every one of the 38 numbers (1 through 36, 0 and 00) has an equally likely chance of showing up. The casino pays off at 35 to 1, so Profit=35 and Risk=1. According to our formula we get 36/38 or 18/19, which would say that for every $19 you put in, the casino gives you $18 in change and asks you nicely if you'd like to play again. In the long run, the casino makes a little more than a nickel on every dollar the players risk at roulette betting single numbers.

Another way to bet is to choose a color, black, red or green. If we choose black, we have 18 of 38 ways to win, 20 of 38 ways to lose (total red + two green) and the payoff is 1 to 1. From the player's view point, this is the same EV, 18/19.

The house understands the two ways of gambling slightly differently, because their risk is different. When a player bets a color, the house only risks $1 for every $1 the player risks, so their EV = 20/19, saying that they see this as about a nickel's worth of profit should come back for every dollar they risk. But when you bet a single number, the house risks $35 when they lose, so the math comes out to 666/665, so the EV comes down to less than a sixth of a penny per roll instead of over a nickel a roll.

This makes the second option seem like a low profit margin for the house, but in reality they still make a nickel per every dollar bet by the players on every spin of the wheel. It's really the money of the losers that pay off the winners in the long run, and the $35 rarely if ever needs to come out of the casino's bank.

The big assumption here is that the wheel is fair, every number equally likely. Back in the 19th Century, a wealthy French gambler hired people to watch the wheels at a Monte Carlo casino spin for more than a day, and when his people reported back to him the observed events, he went to the casino and bet on the wheel that showed the most bias towards a certain number or set of numbers, bet on those numbers and won big. This was immortalized in the popular song The Man Who Broke The Bank at Monte Carlo.


Mathman6293 said...

When I taught middle school math I used to roll 2 weighted dice to determine if I would collect home (I did assign homework for this group)work. The dice had led's and batteries built in so that they'd light up when rolled.

The kids did not realize that these dice actually were biased (I think it was toward 4)due to the internal weights. Naturally, when an even number came is was when I collected.

Distributorcap said...

It's really the money of the losers that pay off the winners in the long run, and the $35 rarely if ever needs to come out of the casino's bank.

same with craps ----

if i am not mistaken, the best bet in the entire casino is behind the line in craps -- the truest odds. the worst odds are at the slots

that is what i am told

and casinos NEVER lose