Posted: 10:35PM Mar 12, 2013||
Assumption 1: "Just visiting" counts as jail.
Assumption 2: If doubles are rolled, then only the final resting place will count.
Both are arbitrary, but had to be assumed one way or the other. I chose this way because I felt like it.
A ten, not doubles, will result in jail instantly. There is a 4/36 or 1/9 chance of this. There is a 1/36 chance of it being a 5-5 pair, so the probability is reduced to 3/36 or 1/12.
A seven will cause a chance card event with a 1/16 chance of jail. The seven has a 1/6 chance of occurring, creating a total probabilty from tens and sevens of 9/96.
A two, P=1/36, can hit jail with P=1/16, creating a 1/576 chance. Added with the previous two yields P=5/576.
I will return soon to do rolls two and three, but I have a headache, it is late, and I'm yet to shower. Alas, I must be exuent at this point.
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Posted: 05:57AM Aug 3, 2013||
'An idea, like a ghost, must be spoken to a little before it will explain itself.' - Charles Dickens
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Posted: 10:25AM Aug 4, 2013||
I'm actually working on this right now. Just so everyone knows I'm actually working on it, here's my calculations so far:
These are the givens I'm using:
Given 1: "Just Visiting" does not count as being IN jail. You must actually land in jail via Go To Jail, or rolling three doubles.
Given 2: Standard dice are used.
Given 3: Get out of jail free card does not affect data, as if you get in jail, you still go to jail.
Given 4: Once you use a card, it gets discarded until all cards in that deck are used.
Given 5: You go first (as the odds change if you don't).
Odds of rolling each total of number:
The odds of rolling doubles are 1/6, as there's only six ways out of 36 possible rolls of getting doubles.
As this is roll 1, the only things that matter are community chest and chance, as the furthest square we can land on is Pall Mall, the twelfth square (for all practical purposes Go counts as square zero). Community chest is square 2, and chance is square 8. The chance of rolling a 2 is 1/36, and the chance of rolling an 8 is 5/36. Chance has a 1/16 chance of going to jail, as does community chest. 1/36 x 1/16 + 5/36 x 1/16 = 1/576 + 5/576 = 6/576, or 1/96 simplified chance of going to jail on first turn after roll 1. However, all other cards in chance have a 5/576 chance of being pulled on roll 1, and the other cards in community chest have a 1/576 chance of getting pulled on roll 1. However, chance has a 1/36 chance of being rolled on doubles, as does community chest. If you land on chance, you have a 16/16 - 7/16 = 9/16 chance of no pertinent card being pulled. If you land on community chest, you have a 16/16 - 3/16 = 13/16 chance of no pertinent card being pulled. Thus, you have a 1/36 x 9/16 = 1/64 chance of landing on and staying on chance with doubles on first roll, and 1/36 x 13/16 = 13/576 chance of landing on community chest and staying on community chest on first roll.
Odds after roll 1 (square you're on is calculated using a roll of doubles on roll 1, being in jail not included in caculation of exclusive double roll, as the square you're on besides jail only matters for calculations if you rolled a double. unfortunately, both chance and community chest are on squares where double roll is possible):
Being on community chest: 13/576
Being on chance: 1/64
Being on go from chance: 1/576
Being on go from community chest: 1/576
Being on Old Kent Road: 1/576
Being on Pall Mall from chance: 1/576
Being on M. Station: 1/576
Being on Trafalgar Square: 1/576
Being on Mayfair: 1/576
Being in jail: 1/96
Being on square 5: 1/576
Being on squares 4, 6, 10, and 12 w/out landing on chance or CC: 1/36 for each of the four squares
This is where it starts getting complicated. We actually have to calculate the odds of landing in jail FROM ALL OF THE POSSIBLE DOUBLE LANDINGS FROM ROLL 1!!!
Starting from community chest (odds = 13/576):
Its not possible to land on the community chest this roll, as the next one is 16 spaces away. However, it is possible to land on chance, if you roll double threes, as its six spaces away. The odds for chance are the same as roll 1, as all cards are in the deck, and six and eight have the same odds of being rolled, and there's only one way to get each double. 1/16 x 5/36 = 5/576 chance of going to jail on roll 2 when you start on community chest. However, you still need to account for the odds of starting from community chest (done in odds section).
Starting from chance (odds = 1/64):
The only way to land on community chest or chance this roll is to roll a ten, which has odds of 3/36. From there, you have a 1/16 chance of landing in jail. That's 1/12 x 1/16 = 1/192 chance of landing in jail this roll. You also have the chances of rolling doubles and going to go from community chest as well as going to Old Kent Road.
Starting from go due to chance (odds = 1/576):
Same odds as roll 1, except chance has 1 less card - the Advance to Go card.
Starting from go due to community chest (odds = 1/576):
Same odds as roll 1, except community chest has 1 less card - the Advance to Go card.
Starting from Old Kent Road (odds = 1/576):
Same odds as roll 1, except community chest is missing a card. However, this doesn't mess up the odds, as there is a zero percent chance of landing on community chest again this turn with doubles (turn, not roll. Its quite impossible to land on chance this roll from doubles, as the only odd spaced chance is 22 spaces away. That means you'd have to land ten or twelve this turn to land on it. And the Go To Jail space? You can forget about landing on that this turn. Go To Jail is space 31. And we can't land on chance or community chest this turn. So the only things that matter when it comes to doubles are doubles, and whether or not those doubles are fives or sixes. However, if you roll a 7, you land on Chance. You still have all Chance cards, so 1/16 x 1/6 = 1/96 chance of landing in jail this roll.
Starting from Pall Mall from chance (odds = 1/576):
The only thing we have to worry about landing on is Community Chest, and pulling the Go To Jail card from Chance, as we passed the only even Chance this turn. the only Community Chest we have to worry about is on square 18, which is six squares away. 5/36 x 1/16 = 5/576 odds of being in jail this roll from Community Chest. Standard odds for the rest of community chest. We have to roll an 11 to land on Chance, so 1/18 x 1/15 = 1/270 odds of landing in jail from Chance. 1/270 + 5/576 = 107/8640 chance of landing in jail this roll (adding all these fractions together is going to be a nightmare!)
Starting from M. Station (odds = 1/576):
The only way we're landing on Chance or Community Chest with doubles is if we roll a two, which will land us on Community Chest. Standard odds for Community Chest, with 1/36 x 1/16 = 1/576 odds of being in jail this roll from Community Chest. You have to roll a seven to get to Chance, so 1/6 x 1/15 = 1/90 chance of landing in jail from Community Chest. 1/576 + 1/90 = 37/2880 chance of landing in jail this roll.
Starting from Trafalgar Square (odds = 1/576):
There is absolutely no way we'll land on Chance or Community Chest this roll with doubles, so we only have to calculate odds of landing in jail. We've alread passed both Chances, so we have to worry about the Go To Jail square and the even-squared Community Chest. Community Chest is nine spaces away, so there's a 1/9 x 1/16 = 1/144 chance of landing in jail this roll. (odds of landing on Go To Jail calculated in odds).
Starting from Mayfair (odds = 1/576):
We're on square 40, but it might as well be square -1 for practical purposes. Funny how the even squares become odd squares when counting backwards. What happened? Square 41 became square 0. The only odd chance is 24 spaces away, so we don't have to worry about landing on that this turn (we'd have to roll double sixes twice in a row, and if that happens, as this is roll 2, we'd go to jail immediately). So, the only things we have to calculate are the odds of landing in jail this roll. 1/18 x 1/16 = 1/288. 1/9 x 1/15 = 1/135. 1/288 + 1/135 = 47/4320 odds of landing in jail this roll.
Starting from square 4 (odds = 1/36):
The next Community Chest is square 18, which is 14 squares away. Not gonna land THERE this roll. However, its possible to land on the Chance on square 8, which is four squares away. That's 1/12 x 1/16 = 1/192 chance of landing in jail this roll, and standard odds for other cards.
Starting from square 6 (odds = 1/36):
We can land on Community Chest OR Chance this roll with doubles! However, the odds of landing on them at all are 1/36 each, as they're 12 and 2 spaces away, respectively. 2(1/36 x 1/16) = 1/36 x 1/8 = 1/288 chance of landing in jail this roll. Standard odds for other cards.
Starting from square 10 (odds = 1/36):
We can't land on Chance this roll, as the next one is 13 spaces away. But we can land on Community Chest, and with doubles, too. Community chest is 8 squares away, so we have a 5/36 x 1/16 = 5/576 chance of landing in jail this roll. Standard odds for other cards.
Starting from square 12 w/out cards pulled (odds = 1/36): Last square to calculate the odds for this roll! (we still have more calculations in the Odds section, though). We can only land on Chance w/out doubles, as Chance is 11 spaces away. 1/18 x 1/16 = 1/288 chance of going to jail with Chance. Community Chest is six spaces away, so 5/36 x 1/16 = 5/576 chance of going to jail from Community Chest this roll. 1/288 + 5/576 = 7/576 odds of going to jail this roll. Standard odds for other cards.
Starting from square 5 (odds = 1/576):
The only pertinent thing we can land on is Chance, as the next Community Chest is 13 squares away. Chance is 3 squares away, so 2/36 x 1/15 = 1/270 chance of going to jail.
Odds for Roll 2:
Landing in Jail: (13/576)(5/576) + (1/64)(1/192) + (1/90)(1/576)(2) + (1/576)(1/96) + (1/576)(107/8640) + (1/576)(37/2880) + (1/576)(1/144) + (1/576)(1/144) + (1/576)(1/4320) + (1/36)(1/192) + (1/36)(1/28 + (1/36)(5/576) + (1/36)(7/576) + (1/270)(1/576) = 65/331776 + 1/12288 + 1/25920 + 1/55296 + 107/4976640 + 37/1658880 + 1/8294 + 1/8294 + 1/2488320 + 1/6912 + 1/10368 + 5/20736 + 7/20736 + 1/15520 = 23/82944 + 47/829440 + 107/4976640 + 982879/6879375360 + 1248307/10319063040 + 5/20736 + 1/1728 = 277/829440 + 107/4976640 + 982879/6879375360 + 1248307/10319063040 + 17/20736 + 1/155520 = 2887/248832 + 107/4976640 + 982879/6879375360 + 1248307/10319063040 = I can't find a calculator that's willing to go further. Gonna have to convert the final odds to decimal format, because I'm NOT finding common denominators for denominators that huge by hand!
"Do not conform to the pattern of this world, but be transformed by the renewing of your mind. Then you will be able to test and approve what God's will is—his good, pleasing and perfect will." -Romans 12:2
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