Thursday, June 28, 2012

Expected Value and why Insurance and Even Money are bad bets

In Mathematics, "Expected Value" is the sum of all the different outcomes of an event multiplied by their respective probabilities. An easy to understand example would be a dice roll. Each face of the die has a 1/6 probability of occurring. So the expected value of a dice roll is:

1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 3.5

This means if you rolled a dice over and over and took the average of those rolls, you'll find that the average approaches 3.5 as the number of rolls increases.

So what does Expected Value (EV) have to do with Blackjack and gambling? Simply put, casinos make their money by having bets and games with negative EV for patrons. Let's take a simple game like roulette. There are 38 different possibilities for the ball to land on (numbers 1-36, 0, and 00). If you were to place $1 on one of those possibilities you can win $35. So 37/38 of the time, you'll be losing your $1 and the other time you'll be winning $35. Let's calculate the EV of our $1 bet:

evRoulette($1) = -$1(37/38) + $35(1/38) = -$0.05263

In the long term, every dollar placed on a game of roulette net's the casino a little more than 5 cents. Multiply this by how many dollars are placed on the table and you can see how casinos make a lot of money.

Let's move on to a more complicated bet: Insurance in Blackjack. Insurance is something that the casino offers when the dealer is showing an Ace. Insurance is basically a side bet that the dealer has Blackjack and it pays 2:1. 2:1 means if I pay $1 on Insurance and the dealer has a Blackjack, I win $2. You usually have a limit of half of your original bet to take insurance. Using our knowledge of the card deck we know that only 4/13 cards give the dealer Blackjack (10, J, Q, K of the 13 different cards) the other 9 values result in a loss of our insurance bet. Let's calculate the EV of a $1 Insurance bet.

evInsurance($1) = -$1(9/13) + $2(4/13) = -$0.07692

Per dollar, this is actually a worse bet than slamming your money down on a random number at routette.

If you've ever sat a Blackjack table you might hear a dealer or some jaded player say, "You ALWAYS take the even money!" Those people are wrong and here is why. Taking even money is getting paid 1:1 on your Blackjack when the dealer is showing an ace. It's 100% chance at getting 1:1 on your original bet. This seems good, right? Calculating the EV of Even Money is simple:

evEvenMoney($1) = $1(1.0) = $1

Well the only way for not taking Even Money to be better is for it to have a higher EV. When you don't take even money, you can either "Push" (meaning you win zero dollars. note: you do not lose your bet) or you get paid 3:2 (1.5:1) on your bet. Let's calculate the EV of not taking Even Money for the same dollar:

evNoEvenMoney($1) = $0(4/13) + $1.5(9/13) = $1.03846

In the long run, not taking even money will be a better bet. Although it may be painful to take a Push on your Blackjack it's actually the profitable choice in the long run.

Seeing these small decimals may not seem like a big deal, but remember that this is per dollar per decision. What happens when I'm betting a little more than a dollar and seeing a lot more hands? Let's look at the Even Money player versus the No Even Money player when they are betting $100 a hand and were dealt Blackjack against an Ace 13 times.

The Even Money Player took even money all 13 times and made a sweet $1300 on his Blackjacks-against-Aces. In those same situations the No Even Money Player never took even money and won $150 nine times and $0 four times netting him a total of $1350. That $50 difference is just in a sample size of 13 hands. If the Blackjack-against-Ace scenario happened 1300 times, the No Even Money Player made $5,000 more than the Even Money Player. It's making the right decision (based on EV) every time that makes the No Even Money player a more profitable player in the long run.

All of this math is a prelude for an upcoming entry that I'm revising for card counting. As you can imagine, EV is everything for card counters.