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Started by webny99, March 08, 2021, 05:58:56 PM

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ilpt4u

#75
Quote from: 1 on March 15, 2021, 07:48:01 PM
Over two turns, it would be 2d12. Over three turns, it would be 3d12.
No, it doesn't

2nd roll is n+1d12. Counting the "total"  combined rolls, even if you roll a 1 on the first roll, you can follow with a 12 and put the 13th spot in play. That is not possible on Roll 1. If you roll a 12 initially, you have a completely new outcome set on your 2nd roll. Everything in-between is a varying probability of having a potential outcome 2 (or more) rolls in a row or an outright new outcome

Also, every throw is an independent event. Assuming a "balanced"  1d12, no individual throw favors any particular number. On a pair of dice, 2d6 or 2d12, you will have an uneven distrubtion of numbers, due to the 2 die not being independent events

The suggestion you make is like saying there is some bell curve to Roulette. Hint: There is not. It is basically a 1d38 die (except that it is a wheel)


Rothman

Just remember, when the event happens, the probability becomes 1. :D
Please note: All comments here represent my own personal opinion and do not reflect the official position(s) of NYSDOT.

hotdogPi

Quote from: ilpt4u on March 15, 2021, 07:57:49 PM
The suggestion you make is like saying there is some bell curve to Roulette. Hint: There is not. It is basically a 1d38 die (except that it is a wheel)

It doesn't matter what the previous number is in roulette. In Monopoly, where you are after two turns is the sum of each turn. Since jail is common, so is 13 spaces past jail (ignoring how doubles to get out of jail works).
Clinched, minus I-93 (I'm missing a few miles and my file is incorrect)

Traveled, plus US 13, 44, and 50, and several state routes

I will be in Burlington VT for the eclipse.

Scott5114

Quote from: 1 on March 15, 2021, 08:05:37 PM
It doesn't matter what the previous number is in roulette. In Monopoly, where you are after two turns is the sum of each turn. Since jail is common, so is 13 spaces past jail (ignoring how doubles to get out of jail works).

That doesn't affect the roll distribution, though.

Look at the chart I posted on the last page: the most frequent roll is 7, because there are more combinations on two six-sided dice that result in a sum of 7 than any other number. Say you are starting from Jail and roll 2d6. Because of the roll distribution,  the most likely space you will land on is 7 spaces after jail, the Community Chest space in the middle of the oranges. The two next most likely spaces to land on are St. James Place (rolling a 6) or Tennessee Avenue (rolling an 8).

If you start on Jail and roll 1d12, each space from St. Charles Place through the Chance space in the middle of the reds is equally likely, because every number 1—12 on a d12 can only be rolled one way (since they each appear as one face of the die), therefore they are equally likely to be rolled.
uncontrollable freak sardine salad chef

hotdogPi

The most commonly landed on property in Monopoly (with the standard 2d6) is 14 spaces past jail – not 6 or 8 (7 isn't a property, but it's still less common than 14 because you can't roll doubles with a 7). You have to look past a single turn.
Clinched, minus I-93 (I'm missing a few miles and my file is incorrect)

Traveled, plus US 13, 44, and 50, and several state routes

I will be in Burlington VT for the eclipse.

Scott5114

Quote from: 1 on March 15, 2021, 08:38:31 PM
The most commonly landed on property in Monopoly (with the standard 2d6) is 14 spaces past jail – not 6 or 8 (7 isn't a property, but it's still less common than 14 because you can't roll doubles with a 7). You have to look past a single turn.

That space–Illinois Avenue–is 7 spaces after the Community Chest space that is 7 spaces after Jail. Thus, it is the result of rolling two sevens in a row (among other things). Or the likeliest next result after already landing on the likeliest space coming out of Jail.

Using 1d12, you only have a 8% chance of landing on that Community Chest space when starting from jail, and you also only have an 8% chance of ending up on Illinois if you do land on the Community Chest space. Using 2d6, you have a 17% chance of landing on the Community Chest space (since 6 of the possible 36 dice rolls land you there), and then a 17% chance of landing on Illinois Avenue after that.
uncontrollable freak sardine salad chef

kphoger

Quote from: Scott5114 on March 15, 2021, 08:49:51 PM

Quote from: 1 on March 15, 2021, 08:38:31 PM
The most commonly landed on property in Monopoly (with the standard 2d6) is 14 spaces past jail – not 6 or 8 (7 isn't a property, but it's still less common than 14 because you can't roll doubles with a 7). You have to look past a single turn.

That space–Illinois Avenue–is 7 spaces after the Community Chest space that is 7 spaces after Jail. Thus, it is the result of rolling two sevens in a row (among other things). Or the likeliest next result after already landing on the likeliest space coming out of Jail.

Using 1d12, you only have a 8% chance of landing on that Community Chest space when starting from jail, and you also only have an 8% chance of ending up on Illinois if you do land on the Community Chest space. Using 2d6, you have a 17% chance of landing on the Community Chest space (since 6 of the possible 36 dice rolls land you there), and then a 17% chance of landing on Illinois Avenue after that.

14  is 7+7 or 6+8 or 8+6.  Isn't that what makes it so common?  (Pressed for time, not working out the math myself right now.)
Keep right except to pass.  Yes.  You.
Visit scenic Orleans County, NY!
Male pronouns, please.

Quote from: Philip K. DickIf you can control the meaning of words, you can control the people who must use them.

hotdogPi

Quote from: Scott5114 on March 15, 2021, 08:49:51 PM
Using 1d12, you only have a 8% chance of landing on that Community Chest space when starting from jail, and you also only have an 8% chance of ending up on Illinois if you do land on the Community Chest space. Using 2d6, you have a 17% chance of landing on the Community Chest space (since 6 of the possible 36 dice rolls land you there), and then a 17% chance of landing on Illinois Avenue after that.

With 1d12, 13 is more common than 14. You have a 1 in 12 chance of landing on 13 past jail after two turns, since you need the correct number for the second roll, and whatever you roll first never makes it impossible. (It's also possible to land on it after 3 or more turns, but that's much rarer.)

Googling 4d6, the chance of rolling 14 is 146/1296, or 11.3%. More of a difference than I thought (1 in 12 is 8.3%), but not the (17%/8%)² that you implied.
Clinched, minus I-93 (I'm missing a few miles and my file is incorrect)

Traveled, plus US 13, 44, and 50, and several state routes

I will be in Burlington VT for the eclipse.

Scott5114

Your understanding of probability is not correct. 2d6 rolled twice is not equivalent to 4d6.

Quote from: kphoger on March 15, 2021, 08:51:20 PM
14  is 7+7 or 6+8 or 8+6.  Isn't that what makes it so common?  (Pressed for time, not working out the math myself right now.)

Yes.
uncontrollable freak sardine salad chef

ilpt4u

I'm loving the 2d6 dice discussion. Getting me warmed up for Vegas Spring Break and the Craps table!

kphoger

Quote from: Scott5114 on March 15, 2021, 08:58:14 PM
Your understanding of probability is not correct. 2d6 rolled twice is not equivalent to 4d6.

Can you explain it to me?  I can't wrap my mind around the statistical difference between the two scenarios:

(a)  Rolling two dice with my right hand and two dice with my left hand, then counting the total number of pips;

(b)  Rolling two dice and counting the number of pips, then rolling two dice again and counting the number of pips.

That is to say, what is wrong with my approach below?




All possible combinations of two rolls, two dice each roll:



Distribution of total pips:

Keep right except to pass.  Yes.  You.
Visit scenic Orleans County, NY!
Male pronouns, please.

Quote from: Philip K. DickIf you can control the meaning of words, you can control the people who must use them.

Scott5114

#86
On 2d6, a roll of 7 has a 6/36 = 17% chance (because there are six ways to roll 7, out of 36 equally likely combinations). On 4d6, a roll of 14 has a 146/1296 = 11% chance of coming up.

(Thanks for doing the combination analysis for me. I was going to do something similar with d4s to reduce the number of combinations, but it got tedious so I stopped.)

The difference between rolling twice as opposed to rolling a combined roll is also important in the context of a game, because something can happen as a result of the first roll that affects the second (for example, you land on a Chance space that moves you to Go or Jail or back 3 spaces, or you land on a space that bankrupts you so there is no way to even do the second roll).
uncontrollable freak sardine salad chef

kphoger

Quote from: Scott5114 on March 15, 2021, 08:58:14 PM
Your understanding of probability is not correct. 2d6 rolled twice is not equivalent to 4d6.

Quote from: kphoger on March 16, 2021, 10:00:18 AM
Can you explain it to me?  I can't wrap my mind around the statistical difference between the two scenarios:

(a)  Rolling two dice with my right hand and two dice with my left hand, then counting the total number of pips;

(b)  Rolling two dice and counting the number of pips, then rolling two dice again and counting the number of pips.

Quote from: Scott5114 on March 16, 2021, 04:48:47 PM
On 2d6, a roll of 7 has a 6/36 = 17% chance (because there are six ways to roll 7, out of 36 equally likely combinations). On 4d6, a roll of 14 has a 146/1296 = 11% chance of coming up.

But we're not only talking about rolling 7 twice.  We're talking about a total of 14 over the course of two rolls, which can be accomplished in 146 ways out of 1296–same as rolling all four dice at the same time.  Right?

Quote from: Scott5114 on March 16, 2021, 04:48:47 PM
The difference between rolling twice as opposed to rolling a combined roll is also important in the context of a game, because something can happen as a result of the first roll that affects the second (for example, you land on a Chance space that moves you to Go or Jail or back 3 spaces, or you land on a space that bankrupts you so there is no way to even do the second roll).

Yes, that will affect the probability in real life, because all spaces are not equal.  But, when it comes to the actual rolling of the dice, I still don't see the difference.
Keep right except to pass.  Yes.  You.
Visit scenic Orleans County, NY!
Male pronouns, please.

Quote from: Philip K. DickIf you can control the meaning of words, you can control the people who must use them.

Scott5114

At this point I would have to consult the experts who taught me this to be confident in not getting details wrong going forward. Such is the problem with math.
uncontrollable freak sardine salad chef

NWI_Irish96

Quote from: kphoger on March 16, 2021, 05:05:10 PM
Quote from: Scott5114 on March 16, 2021, 04:48:47 PM
The difference between rolling twice as opposed to rolling a combined roll is also important in the context of a game, because something can happen as a result of the first roll that affects the second (for example, you land on a Chance space that moves you to Go or Jail or back 3 spaces, or you land on a space that bankrupts you so there is no way to even do the second roll).

Yes, that will affect the probability in real life, because all spaces are not equal.  But, when it comes to the actual rolling of the dice, I still don't see the difference.

Yeah, unless there is some kind of intermediate outcome based on the first of two rolls, rolling a pair of dice twice produces the same outcome probabilities as rolling four dice once.
Indiana: counties 100%, highways 100%
Illinois: counties 100%, highways 61%
Michigan: counties 100%, highways 56%
Wisconsin: counties 86%, highways 23%

Big John


hotdogPi

#91
Quote from: Big John on March 21, 2021, 04:20:55 PM
Monopoly to change "outdated" Community Chest cards: https://www.radio.com/news/monopoly-replaces-outdated-community-chest-cards-in-makeover

"You have invented a time machine. Keep this card. After rolling, before you move, you may put it on the bottom of the Community Chest pile and reroll the dice. This card may be sold or traded."

"Your GPS hasn't been updated and you take a wrong turn. You must switch places with another player. Act as if you landed on the new space you are on, but the player you swapped with does not draw a Community Chest card."

"Your car is having problems. For the next two rolls, roll one die instead of two."

"Stimulus check! If you have less than $1,000 and don't own any monopolies, collect $600."

"Crap! The next player to roll a 7 must pay $100."
Clinched, minus I-93 (I'm missing a few miles and my file is incorrect)

Traveled, plus US 13, 44, and 50, and several state routes

I will be in Burlington VT for the eclipse.

formulanone

#92
Quote from: Big John on March 21, 2021, 04:20:55 PM
Monopoly to change "outdated" Community Chest cards: https://www.radio.com/news/monopoly-replaces-outdated-community-chest-cards-in-makeover

But...finishing in second place in that beauty contest is the unexpected reward for something nobody thought they were contesting. It's only a diss if you're vain on a level of children's stories, but a promotion for just being yourself. It's a reward because your mother always thinks you look good. It's the complement you never knew you needed. And even though it won't save you much when landing on someone else's hotel, it's that extra mint on your pillow.

One really has to be lacking antidepressants to find a negative meaning for that card. At no time does the card ask anyone to bear their chest, make it rain one-sided bills, or throw the prettiest player into the volcano* for the card to have any deleterious effect on morale and gender stereotypes.



* house rules may vary.

Scott5114

How do you even end up in a beauty contest without entering? Were unexpected beauty contests, where everyone in some store or whatever would be rounded up and you might get given $10 for being prettier than everyone but one person, a thing back in the 1930s when the game was made? And if not, would someone enter a beauty contest if they didn't think they'd get first prize?
uncontrollable freak sardine salad chef

1995hoo

"You know, you never have a guaranteed spot until you have a spot guaranteed."
—Olaf Kolzig, as quoted in the Washington Times on March 28, 2003,
commenting on the Capitals clinching a playoff spot.

"That sounded stupid, didn't it?"
—Kolzig, to the same reporter a few seconds later.

hotdogPi

Quote from: 1995hoo on April 14, 2021, 08:39:36 AM


Look carefully. The closest side of the game board to the reader has 8 spaces between corners. The actual Monopoly board has 10.
Clinched, minus I-93 (I'm missing a few miles and my file is incorrect)

Traveled, plus US 13, 44, and 50, and several state routes

I will be in Burlington VT for the eclipse.

kphoger

Quote from: 1 on April 14, 2021, 08:49:50 AM

Quote from: 1995hoo on April 14, 2021, 08:39:36 AM




Look carefully. The closest side of the game board to the reader has 8 spaces between corners. The actual Monopoly board has 10.

Look carefully.  The kids don't have the correct number of fingers.

Look carefully.  The dad's nose is disproportionately large for his face.

Look carefully.  The mom's eyes go nearly all the way to the top of her skull.

Look carefully.  The boy is wearing Calvin's shirt.
Keep right except to pass.  Yes.  You.
Visit scenic Orleans County, NY!
Male pronouns, please.

Quote from: Philip K. DickIf you can control the meaning of words, you can control the people who must use them.

Scott5114

Quote from: kphoger on April 14, 2021, 10:25:15 AM
Quote from: 1 on April 14, 2021, 08:49:50 AM

Quote from: 1995hoo on April 14, 2021, 08:39:36 AM




Look carefully. The closest side of the game board to the reader has 8 spaces between corners. The actual Monopoly board has 10.

Look carefully.  The kids don't have the correct number of fingers.

Look carefully.  The dad's nose is disproportionately large for his face.

Look carefully.  The mom's eyes go nearly all the way to the top of her skull.

Look carefully.  The boy is wearing Calvin's shirt.

Look carefully. The mom is a cyclops whose eyeball has two pupils. (Perhaps she's distantly related to Sonic the Hedgehog.)
uncontrollable freak sardine salad chef

I-55

Quote from: 1 on March 21, 2021, 04:27:27 PM
Quote from: Big John on March 21, 2021, 04:20:55 PM
Monopoly to change "outdated" Community Chest cards: https://www.radio.com/news/monopoly-replaces-outdated-community-chest-cards-in-makeover

"You have invented a time machine. Keep this card. After rolling, before you move, you may put it on the bottom of the Community Chest pile and reroll the dice. This card may be sold or traded."

"Your GPS hasn't been updated and you take a wrong turn. You must switch places with another player. Act as if you landed on the new space you are on, but the player you swapped with does not draw a Community Chest card."

"Your car is having problems. For the next two rolls, roll one die instead of two."

"Stimulus check! If you have less than $1,000 and don't own any monopolies, collect $600."

"Crap! The next player to roll a 7 must pay $100."

"Trains have been delayed! Each player owes $100 per railroad owned."

"It snowed in Texas! If you own any utilities, pay $250"

"Your wife is divorcing you! Give everything to the bank, you lose!"
Let's Go Purdue Basketball Whoosh

Ned Weasel

My favorite game is played on mats, and it's called wrestling.

My favorite table-top game is arm-wrestling.

As for board games, about the only two that I've ever been able to get interested in are Chess and Monopoly.  I did read some of the earlier discussion about how people can exploit Monopoly because of the dice system.  I hadn't given it that much thought before.  My personal strategy is usually to buy as many properties as feasible up-front, and then build on the cheaper properties as quickly as possible.  I've known a fair amount of people who are really good at Chess, and I think it's because they have more patience than I do.
"I was raised by a cup of coffee." - Strong Bad imitating Homsar

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