My friend recently went to a casino to play craps. He showed me the table and it got me thinking about odds. With my calculations below I have calculated the casino losing money, which is simply not possible..so where did I go wrong with my calculations?
FYI, I know very little about probability/stats, so my calculations/assumptions are probably completely wrong.
The craps table at the casino pays 30 to 1 for rolling 2 1's or 2 6's. I decided to calculate the odds:
- 2 Dice = 36 different rolls
- Probability of rolling two 1's: 1/36
- Probability of rolling two 6's: 1/36
- Total probability: 2/36 = 5.5%
I assume probability works better over large numbers, so let's say I bet 5$ over 100 bets.
Over 100 bets I would've spent $500.
However, I should've won, at least 5 times. Thats
$5bet*30xmultipler =$150per win.$150per win*5wins =$750.
$750 in winnings - $500 cost = $250 (Actually cost is less since 5 of the rounds I won and it didn't cost me anything for those rounds, but for simplicity sakes)
Did I just win $250? Are my calculations correct?
If you have a bet that pays $x$ to $1$ and the chances of winning are $1$ out of $y$, then in expectation you are making: $$+x\cdot \frac{1}{y} - 1\cdot\frac{y-1}{y}$$you have a positive expectation when the above is greater or equal to $0$, so $$\frac{x}{y} - \frac{y-1}{y} \geq 0$$ which gives $x+1 \geq y$. Using your numbers, you have $x = 30$ and $y = 18$, so it is worth it to play.
Hence,
1) Those aren't the odds and you miss read them. Or,
2) The casino is going to go bust sometime soon.