odiousgambit
odiousgambit
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August 22nd, 2020 at 4:11:09 AM permalink
can someone distill for me why integral calculus is needed? to my 'gut' it just doesn't seem like a Craps analysis would ever require it
the next time Dame Fortune toys with your heart, your soul and your wallet, raise your glass and praise her thus: “Thanks for nothing, you cold-hearted, evil, damnable, nefarious, low-life, malicious monster from Hell!” She is, after all, stone deaf. ... Arnold Snyder
Ace2
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Thanks for this post from:
odiousgambit
August 22nd, 2020 at 8:17:18 AM permalink
Quote: odiousgambit

can someone distill for me why integral calculus is needed? to my 'gut' it just doesn't seem like a Craps analysis would ever require it

It’s not actually required, but you’d have to algebraically work through a staggering amount of states otherwise.

Here’s an example. There’s a 6 x 6 checkerboard and 6 checkers. In the first five columns there’s a checker in the first row and in the six column the checker is in the second row. We will roll a standard die...if a 4 is rolled we move the checker in column 4 up 1 row. We continue rolling until one of the checkers makes it to row 6. What is the probability that the checker in column 6 (with a 1 row head start) wins? Simple dice problem, right?

Solving this problem is quite similar to solving the Replay Bet since it’s about figuring which outcome will accumulate a certain amount of “wins” first. In other words, it’s a race, which is often ideal for a Poisson integration.

For the checkerboard problem you could list out all 5^5 * 4 = 12,500 states and tediously work through them with algebra. Or you could solve it quickly by integrating one short equation.

Extra credit: what’s the answer to the checkerboard problem?
Last edited by: Ace2 on Aug 22, 2020
It’s all about making that GTA
Wizard
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August 22nd, 2020 at 9:43:59 AM permalink
Quote: Steen

Nice job Michael!



Thank you!
It's not whether you win or lose; it's whether or not you had a good bet.
Wizard
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August 22nd, 2020 at 9:48:02 AM permalink
Quote: odiousgambit

can someone distill for me why integral calculus is needed? to my 'gut' it just doesn't seem like a Craps analysis would ever require it



You could do a Markov chain, but there are 5^6=15,625 states to the game. You would have to determine the probability of each state leading to each state. Very tedious.

I find it amazing that calculus can be used to solve a discrete problem like that.
It's not whether you win or lose; it's whether or not you had a good bet.
Wizard
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August 22nd, 2020 at 10:26:18 AM permalink
Quote: Ace2

Here’s an example. There’s a 6 x 6 checkerboard and 6 checkers. In the first five columns there’s a checker in the first row and in the six column the checker is in the second row. We will roll a standard die...if a 4 is rolled we move the checker in column 4 up 1 row. We continue rolling until one of the checkers makes it to row 6. What is the probability that the checker in column 6 (with a 1 row head start) wins? Simple dice problem, right?

Solving this problem is quite similar to solving the Replay Bet since it’s about figuring which outcome will accumulate a certain amount of “wins” first. In other words, it’s a race, which is often ideal for a Poisson integration.

For the checkerboard problem you could list out all 5^5 * 4 = 12,500 states and tediously work through them with algebra. Or you could solve it quickly by integrating one short equation.

Extra credit: what’s the answer to the checkerboard problem?




5426820388378871923714339111 / 16736565124800000000000000000 = Approximation: 0.3242493515194159



Integrate for x from 0 to infinity of (1-exp(-x/6)*(1+(x/6)+(x/6)^2/2+(x/6)^3/6))*(exp(-x/6)*(1+(x/6)+(x/6)^2/2+(x/6)^3/6+(x/6)^4/24))^5/6.

The function in the integral represents the probability at the moment the game ends the checker in column 6 will have advanced four or more times AND the other five checkers will have advanced 0 to 4 times each.
It's not whether you win or lose; it's whether or not you had a good bet.
TDVegas
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August 22nd, 2020 at 10:34:32 AM permalink
You’re better off playing the Britney Spears slots.
Ace2
Ace2
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August 22nd, 2020 at 11:18:17 AM permalink
Quote: Wizard


5426820388378871923714339111 / 16736565124800000000000000000 = Approximation: 0.3242493515194159



Integrate for x from 0 to infinity of (1-exp(-x/6)*(1+(x/6)+(x/6)^2/2+(x/6)^3/6))*(exp(-x/6)*(1+(x/6)+(x/6)^2/2+(x/6)^3/6+(x/6)^4/24))^5/6.

The function in the integral represents the probability at the moment the game ends the checker in column 6 will have advanced four or more times AND the other five checkers will have advanced 0 to 4 times each.

I disagree.

I believe you’ve calculated the probability that, at any given time:

1) column 6 has advanced >3
2) columns 1-5 have advanced <5
3) column 6 wins next roll

If column 6 has advanced >3, it’s already won and doesn’t need another win. Also, it can’t continue winning at every position above the fifth row, as your formula implies
It’s all about making that GTA
Wizard
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August 22nd, 2020 at 12:21:13 PM permalink
Quote: Ace2

I disagree.



I agree. I was thinking about this away from the computer and had another idea. Let me try again.

6564030407855551 / 21936950640377856

Approximation: 0.2992225544681486


Integrate for x from 0 to infinity of:

exp(-x/6)*(x/6)^3/6*(exp(-x/6)*(1+(x/6)+(x/6)^2/2+(x/6)^3/6+(x/6)^4/24))^5*(1/6)

This should be the probability of the checker in column 6 advancing 3 more squares, the other five advancing 0 to 4, and the one in column 6 advancing next (and winning).
Last edited by: Wizard on Aug 22, 2020
It's not whether you win or lose; it's whether or not you had a good bet.
Ace2
Ace2
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August 22nd, 2020 at 12:26:26 PM permalink
Quote: Wizard

I agree. I was thinking about this away from the computer and had another idea. Let me try again.

6564030407855551 / 21936950640377856

Approximation: 0.2992225544681486


Integrate for x from 0 to infinity of:

exp(-x/6)*(x/6)^3/6*(exp(-x/6)*(1+(x/6)+(x/6)^2/2+(x/6)^3/6+(x/6)^4/24))^5*(1/6)

This should be the probability of the checker in column 6 advancing 3 more squares, the other five advancing 0 to 4, and the one in column x advancing next (and winning).

Well done Wizard. I owe you an O’Douls
It’s all about making that GTA
Wizard
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August 22nd, 2020 at 12:43:21 PM permalink
Quote: Ace2

Well done Wizard. I owe you an O’Douls



Thanks Ace! My first time winning a drink, if you can call O'Douls that.

I went to a soccer game in Barcelona and thought I was ordering a beer. Little did I know until after standing in a long line and taking a sip all they sold was non-alcoholic beer. *sheesh*
It's not whether you win or lose; it's whether or not you had a good bet.

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