# The best Roulette play (by expected value)

Date: 2023-02-18 | finance

I'm in Vegas for the weekend cause Megna had a sales conference so I had a free hotel. It's my first time here - I'm not a big gambler / clubber - but when in Vegas.

I enjoy running expected values on gambling games cause it's remarkable how large a legal scam it all is. But that's what makes it fun I guess.

So in this post I wanted to do some rough math to figure out what the best play for Roulette is. The idea is that if you can lose the least amount of money possible while playing Roulette for the longest, you may be able to get enough drinks from the hosts to break evenish.

# Math Overview

I'm just gonna use some basic expected value maths. This is basically SUM(allOutComeValues) where each outcomeValue is every possible outcome, its value return * its odds of occurring.

So:

- expectedValue = oddsOfOccuring * resultOutcome

I'm doing this on "American" Roulette (38 numbers) and pulling data on Roulette payouts from The Venetian.

# Roulette Plays and Expected Values

To calculate for roulette itself, we can do:

- Expected Value = Win + Loss
- Win = Odds * Payout * Bet
- Loss = Odds * -BET

To keep things simple, we'll just assume that the bet is $1 so you can multiply the outcome by your actual bet to see how much you're losing.

Possible Plays:

- Single Number: -$0.053
- Pays: 35:1
- Odds: 1/38 = 0.026 = 2.6%
- Expected Value: -0.053
- Win = (1/38) * 35 = 0.921
- Loss = (37/38) * -1 = -0.974

- Double Number: -$0.052 -> -$0.053
- Pays: 17:1
- Odds: 2/38 = 0.053 = 5.3%
- Expected Value: -0.052
- Win = (2/38) * 17 = 0.895
- Loss = (36/38) * -1 = -0.947

- Three Number: -$0.053
- Pays: 11:1
- Odds: 3/38 = 0.0789 = 7.9%
- Expected Value: -0.053
- Win = (3/38) * 11 = 0.868
- Loss = (35/38) * -1 = -0.921

- Four Number: -$0.053
- Pays: 8:1
- Odds: 4/38 = 0.105 = 10.5%
- Expected Value: -$0.053
- Win = (4/38) * 8 = 0.842
- Loss = (34/38) * -1 = -0.895

- Five Number: -$0.079
- Pays: 6:1
- Odds: 5/38 = 0.132 = 13.2%
- Expected Value: -$0.079
- Win = (5/38) * 6 = 0.789
- Loss = (33/38) * -1 = -0.868

- Note: This is a special bet for 0-00-1-2-3

- Six Number: -$0.053
- Pays: 5:1
- Odds: 6/38 = 0.158 = 15.8%
- Expected Value: -$0.053
- Win = (6/38) * 5 = 0.789
- Loss = (32/38) * -1 = -0.842

- Twelve Number / column: -$0.053
- Pays: 2:1
- Odds: 12/38 = 0.316 = 31.6%
- Expected Value: -$0.052 -> -$0.053
- Win = (12/38) * 2 = 0.632
- Loss = (26/38) * -1 = -0.684

- 18 numbers / red / black / odds / evens: -$0.053
- Pays: 1:1
- Odds: 18/38 = 0.474 = 47.4%
- Expected Value: -$0.052 -> -$0.053
- Win = (18/38) * 1 = 0.474
- Loss = (20/38) * -1 = -0.526

This tends to agree with the idea that the house edge is 2/38 = 0.053 = 5.3% (Wikipedia). So for every $1 you bet, you're expected to lose 5.3 cents of it.

# The Best Roulette Play

So what's the best roulette play?

The truth is that they all suck. The house has rigged the game so that every play you lose an average of 5.3% of your bet.

But we're here to have fun so the answer is: any play is equally bad at -5.3% (except for the 5 play which is worse at -7.9%).

So play however you want, but keep your bets low and slow so you can get as many "free" drinks as you can. =)