Rolling a 100-sided die
April 18th, 2019 at 4:55:12 PM
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Since Wiz added a math problem, I thought I would as well - but mine deals with probability.
(Unashamedly "borrowed" from this year's MIT high school mathematics tournament's combinatorics round - 10 questions, 50 minutes)
A casino offers the following game:
You place a bet, and start with 0 points.
You roll a 100-sided die that has the numbers 0, 1, 2, ..., 98, 99 on it, 100 times.
Each time you roll, if you roll less than or equal to your current score, you gain a point.
After the 100 rolls, you are paid back (your bet multipled by your score divided by 2).
In other words, if you bet $10 and your score is 1, you lose $5; if it is 2, you break even; if it is 3, you gain $5; and so on.
What's the house edge?
(Unashamedly "borrowed" from this year's MIT high school mathematics tournament's combinatorics round - 10 questions, 50 minutes)
A casino offers the following game:
You place a bet, and start with 0 points.
You roll a 100-sided die that has the numbers 0, 1, 2, ..., 98, 99 on it, 100 times.
Each time you roll, if you roll less than or equal to your current score, you gain a point.
After the 100 rolls, you are paid back (your bet multipled by your score divided by 2).
In other words, if you bet $10 and your score is 1, you lose $5; if it is 2, you break even; if it is 3, you gain $5; and so on.
What's the house edge?
April 18th, 2019 at 6:33:26 PM
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does it matter that there are no fair 100 sided dice?Quote: ThatDonGuySince Wiz added a math problem, I thought I would as well - but mine deals with probability.
(Unashamedly "borrowed" from this year's MIT high school mathematics tournament's combinatorics round - 10 questions, 50 minutes)
A casino offers the following game:
You place a bet, and start with 0 points.
You roll a 100-sided die that has the numbers 0, 1, 2, ..., 98, 99 on it, 100 times.
Each time you roll, if you roll less than or equal to your current score, you gain a point.
After the 100 rolls, you are paid back (your bet multipled by your score divided by 2).
In other words, if you bet $10 and your score is 1, you lose $5; if it is 2, you break even; if it is 3, you gain $5; and so on.
What's the house edge?
Climate Casino: https://climatecasino.net/climate-casino/
April 18th, 2019 at 7:15:47 PM
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$315.14
I used a spreadsheet. I'm not immediately sure how I would solve it without one.
I used a spreadsheet. I'm not immediately sure how I would solve it without one.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
April 18th, 2019 at 10:14:13 PM
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Quote: ThatDonGuy...What's the house edge?
I could never solve it with pencil and paper, but with mouse and Excel I got a house edge of 0.147 593 085 289 237.
April 18th, 2019 at 10:31:01 PM
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Quote: Wizard$315.14
I used a spreadsheet. I'm not immediately sure how I would solve it without one.
How is that a house edge? O _ O
It’d probably take me a few days to think about even trying to figure out how to solve this problem with pen & paper (pencils suck).
April 18th, 2019 at 10:37:56 PM
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Quote: Wizard$315.14
I used a spreadsheet. I'm not immediately sure how I would solve it without one.
Does that mean the house edge is 68.5% ?
April 18th, 2019 at 11:40:47 PM
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Quote: ThatDonGuy...What's the house edge?
I guess the average score after the nth roll would be [the average score after the (n-1)th roll] plus [1+ average score after the (n-1)th roll]/100.
After some algebraic manipulation with pencil and paper, I get an average final score of 1.01^100 - 1. This would be a return of 0.5 *(1.01^100-1) = 0.8524 which means a house edge of 0.1476.
After some algebraic manipulation with pencil and paper, I get an average final score of 1.01^100 - 1. This would be a return of 0.5 *(1.01^100-1) = 0.8524 which means a house edge of 0.1476.
April 19th, 2019 at 5:09:12 AM
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Quote: ChesterDogI guess the average score after the nth roll would be [the average score after the (n-1)th roll] plus [1+ average score after the (n-1)th roll]/100.
After some algebraic manipulation with pencil and paper, I get an average final score of 1.01^100 - 1. This would be a return of 0.5 *(1.01^100-1) = 0.8524 which means a house edge of 0.1476.
Clever!
Amazing to think a high school student gets there in 5 minutes . . .
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
April 19th, 2019 at 6:45:42 AM
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Quote: RSHow is that a house edge? O _ O
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
April 19th, 2019 at 6:59:25 AM
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Sorry, I misread the problem initially and gave the player a point for rolling equal or more than his current score. Then I just posted the expected win, as opposed to a house edge. I owe the forum two punishments for that.
house edge = 16.0983253%
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
April 19th, 2019 at 7:39:21 AM
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Using a brute force (100x100) spreadsheet I get the same answer. Essentially at each roll I tried to work out the chances of being on each of the scores 0 thru 100. btw - Wizard I get your figure by looking at what happens after 99 rolls.Quote: ChesterDog...a house edge of 0.1476...
Using the same idea I got 14.154% for a 1000-sided dice, suggesting it approaches a nice number as N tends to infinity
April 19th, 2019 at 4:07:45 PM
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Quote: charliepatrickWizard I get your figure by looking at what happens after 99 rolls.
D'oh! You're right.
0.147593085
Last edited by: Wizard on Apr 20, 2019
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
April 19th, 2019 at 4:25:32 PM
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Quote: ChesterDogI guess the average score after the nth roll would be [the average score after the (n-1)th roll] plus [1+ average score after the (n-1)th roll]/100.
After some algebraic manipulation with pencil and paper, I get an average final score of 1.01^100 - 1. This would be a return of 0.5 *(1.01^100-1) = 0.8524 which means a house edge of 0.1476.
Correct - the expected number of wins is 1.01100 - 1, which results in a house edge of about 14.76%
