Coder, data scientist and maths enthusiast.

# How to place a positive expectation bet in roulette

If you’ve ever taken a basic statistics class you’re first reaction is probably “you can’t”. To a degree you’re right. To a more accurate degree you’re wrong. You certainly can’t beat the house. You can invent any betting system you like, it doesn’t matter. All it does it dictate when you’re going to lose the money. However you can place a positive expectation bet.

The key is to consider what you would do with the winnings. If you would invest them you should factor this into the expectation. Having more money opens up new investment options. The simple and unfortunate fact is that the more money you have the easier it is to get more.

Obviously this applies anywhere where it will open up a new investment opportunity. It also applies to any wager, not just roulette. However in this article I will work through an example with roulette to show that most people are capable of placing a positive expectation bet. Even though they might not want to.

So lets say you’re young and trying to save up to buy a house (or are old and want another house). You need more money to put down a deposit for a mortgage and buy the house. Assume that you will spend a constant amount of money on a house with a mortgage as soon as you have enough for a deposit. All you have to do now is get the money for the deposit. Obviously the sooner you get the money, the better.

Here I’ll work through a plausible example of this - betting £1000 on a single number of the roulette wheel to try and win a deposit for a house.

#### Time

Lets say for arguments sake that deposit is going to be £36,000 and you’re saving £1000 a month. It’s going to take you 36 months or 3 years to earn that deposit. Of course if you bet £1000 on 13, black it would take you either 1 month or 37 months, depending on if you win.

#### Rent

So how much would you gain or lose due to throwing it into the black hole of rent? Lets say that rent is costing you £400 a month. If you don’t bet, you’ll spend £14,400 on rent in those three years. If you win the bet you’ll spend £400 and you’ll spend £14,800 if you lose.

So although the payout of roulette is 36/1 the actual chance of winning (with a double zero) is $\frac{1}{38}$ . So if everyone did this the average rent paid would be: $\frac{1}{38}*400+\frac{37}{38}*14800$ , which is £14,421.05

If you didn’t place the bet, you would pay £14,400. So you expect to pay £21.05 more if you place the bet. That’s not much though. What if there’s another factor that will dominate it. Something like:

#### Rising house prices

We’re going to work on the assumption that house prices are going to continue their current projected rise in price. The population is going up and people need places to live. So this feels like a relatively safe assumption. The ONS give 5.4% as the increase for 2013 and 11.8% for 2014. Projections for 2015 are a little more pessimistic, guessing at about 3%.

So lets go with the projected 3% as it’s the most pessimistic. Normally that money would go into a savings account, which would get you about 1.4% at the moment, so we’ll take that off. To give us an extra 1.6% pa on our house price. If again for the sake of argument we are planning to buy a house worth £120,000 if we get the house two years, eleven months early we will make an extra £5,686.29 from rising house prices.

#### Putting it all together

So the additional expectation you can expect from the rising house prices is: $\frac{1}{38}*5686.29$

Which comes to £149.64. If you made the bet you would expect to pay an extra £21.05 in rent. So that means a positive expectation of £128.59 for placing the bet.

So there you go. It is possible to have a positive expectation from a bet on roulette. So mathematically speaking it’s something you should probably do. Sanely speaking it’s probably too much risk for most people.

There’s other factors, such as assumptions we’ve made. The housing market might crash again, like it did in 2008. The change in house price figure was somewhat plucked out of the air. Though no matter what that figure is (so long as it’s above what you get in a savings account), with low enough rent it will still have a positive expectation. So it doesn’t invalidate the point of this article.

Things get much more complicated when you try and work out what a good deposit for a house would be. Too little and you’re going to pay incredible amounts in interest. When the right time to buy is from that perspective will get very complicated.

(I accept no responsibility for any horrible financial decisions you make on account of reading anything you’ve just read.)