---
title: 'The Mathematics of Roulette I Understanding Casino Games'
source: 'https://youtube.com/watch?v=pYKcPL0N9QQ'
video_id: 'pYKcPL0N9QQ'
date: 2026-07-06
duration_sec: 0
---

# The Mathematics of Roulette I Understanding Casino Games

> Source: [The Mathematics of Roulette I Understanding Casino Games](https://youtube.com/watch?v=pYKcPL0N9QQ)

## Summary

This lecture explains the concept of expected value using American roulette as an example. It shows how to calculate the expected loss per bet, which is the same for nearly all bets at 5.3 cents per dollar.

### Key Points

- **Roulette wheel composition** [00:05] — American roulette has 38 numbers: 18 red, 18 black, and 2 green (0 and 00). European roulette has one green number.
- **Probability of winning on red** [00:51] — The probability of winning a bet on red is 18/38, slightly less than 50%.
- **Expected value definition** [01:18] — Expected value is a weighted average of possible outcomes. For a $1 bet on red, expected value = (1 * 18/38) + (-1 * 20/38) = -2/38 ≈ -$0.0526.
- **Bet on numbers 1-12** [02:16] — Pays 2 to 1. Expected value = (2 * 12/38) + (-1 * 26/38) = -2/38 ≈ -$0.0526.
- **Bet on a single number** [03:39] — Pays 35 to 1. Expected value = (35 * 1/38) + (-1 * 37/38) = -2/38 ≈ -$0.0526.
- **Consistent expected loss** [04:24] — Nearly every bet in roulette has the same expected value of -5.3 cents per dollar bet.

### Conclusion

In American roulette, the expected value for nearly all bets is -5.3 cents per dollar, illustrating the house edge.

## Transcript

these numbers are red 18 of these numbers are black and two of the numbers zero and double zero are green by the way i'm describing american roulette the european version has just one green number
the simplest bet in roulette is to bet on one of the main colors let's say red red it's an even money bet which means that if you bet a dollar then you'll either win or lose one dollar depending on
whether or not a red number appears here let's give it a try here
since there are 38 numbers each of which has the same chance of occurring and 18 of these numbers are read then the probability that you win is 18 over 38
which is a little less than 50 percent clearly you have a disadvantage at this game we can quantify this disadvantage using the very important concept of expected
value if you only remember one concept from this lecture this is what i want you to remember the expected value of a bet is a the expected value of a bet is a weighted average of how much you can win
or lose when you bet on red in roulette you'll either win a dollar with probability 18 over 38 or you'll lose a dollar or you could say win negative one dollars with probability 20 over 38 right because
there are 18 red numbers and 20 numbers that aren't hence your expected value is 1 times 18 over 38 plus negative 1
is 1 times 18 over 38 plus negative 1 times 20 over 38 that's negative 2 over times 20 over 38 that's negative 2 over 38 negative 0.0526 what this means is that on average you'll lose about 5.3 cents for every
dollar that you bet now in roulette you can bet on other things besides color for instance you can bet that a number between 1 and 12 shows up here let me show you here so let's say we bet a dollar that one of
the first 12 numbers shows up the casino pays two to one odds for this bet which means that if you bet a dollar and you win then the casino pays you two
so let's calculate our expected value here so when you bet a dollar you're going to win win two dollars with probability 12 numbers out of 38
and you're going to lose a dollar with probability 26 over 38 because if you win 12 times 26 over 38 because if you win 12 times out of 38 you lose 26 times out of 38.
hence the expected value of this bet is is 2 that's what you win 12 over 38 times 2 that's what you win 12 over 38 times plus negative 1 that's for losing 1 26
out of 38 times when you do the math that's negative 2 when you do the math that's negative 2 over 38 negative 0.0526 which is the same number as before or suppose you bet on a single number
let's say i like lucky number 17. all right here the casino pays 35 to 1 odds thus when you make this bet then you either win 35 with probability 1 over 38 just one
winning number out of all 38 or you lose one dollar with probability 37 over 38. so when you calculate the expected value 35 times 1 over 38 plus
expected value 35 times 1 over 38 plus negative 1 times 37 over 38 once again negative 1 times 37 over 38 once again you get negative 2 over 38. we still get negative 5.3 cents interestingly when you play roulette
practically every bet has the exact same expected value of negative 5.3 cents per dollar bet [Music]