Roulette Bets
How To Find The House Edge For Any Roulette Bet
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Let B |
= |
amount bet |
| |
W |
= |
amount you stand to win |
| |
y |
= |
payoff odds =
amount won per dollar bet
[ so that W = yB ]
|
| |
p |
= |
probability of winning |
| |
| |
E |
= |
expectation |
| |
= |
expected value of W |
| |
= |
Wp + ( - B ) * ( 1 - p ) |
| |
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Replacing W by yB, we get |
| |
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E |
= |
yBp + ( - B ) * ( 1 - p )
= yBp
- B + Bp
= B * ( yp - 1 + p )
|
| |
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E / B |
= |
-1 + p( y+1 ) |
| |
| |
H |
= |
House Edge |
| |
= |
| E / B | |
| |
= |
1 - p ( y + 1 ) |
| |
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The value of y depends on the type of bet,
and casinos use these values:
|
| |
Bet on red or black |
→
|
y = 1 |
| |
Bet on a single number |
→
|
y = 35 |
| |
Bet on two numbers |
→
|
y = 17 |
| |
Bet on three numbers |
→
|
y = 11 |
| |
Bet on four numbers |
→
|
y = 8 |
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Bet on five numbers |
→
|
y = 6 |
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Bet on six numbers |
→
|
y = 5 |
| |
Bet on twelve numbers |
→
|
y = 2 |
For the "Five Number" bet,
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H |
= |
1 - p ( y + 1 )
|
| |
= |
1 - ( 5 / 38 )( 7 )
|
| |
= |
1 - ( 35 / 38 )
|
| |
= |
3 / 38
|
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» |
7.89% |
For any "two number" bet,
| |
H |
= |
1 - p ( y + 1 )
|
| |
= |
1 - ( 2 / 38 )( 18 )
|
| |
= |
1 / 19
|
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» |
5.26% |
You can verify the following fact:
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For all bets other than the five-number bet,
H = 1 / 19.
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