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Message boards : General discussion : Probability of small straight in yahtzee

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Discovered 8 mega primesFound 5 primes in the 2020 Tour de PrimesFound 2 mega primes in the 2020 Tour de PrimesFound 4 primes in the 2021 Tour de PrimesFound 2 mega primes in the 2021 Tour de Primes321 LLR Sapphire: Earned 20,000,000 credits (20,020,701)Cullen LLR Sapphire: Earned 20,000,000 credits (20,174,451)ESP LLR Sapphire: Earned 20,000,000 credits (23,496,803)Generalized Cullen/Woodall LLR Sapphire: Earned 20,000,000 credits (20,197,125)PPS LLR Emerald: Earned 50,000,000 credits (61,145,268)PSP LLR Sapphire: Earned 20,000,000 credits (22,678,721)SoB LLR Sapphire: Earned 20,000,000 credits (26,978,901)SR5 LLR Sapphire: Earned 20,000,000 credits (20,183,763)SGS LLR Sapphire: Earned 20,000,000 credits (20,482,069)TRP LLR Sapphire: Earned 20,000,000 credits (20,307,419)Woodall LLR Sapphire: Earned 20,000,000 credits (22,371,625)321 Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,380,527)Cullen/Woodall Sieve (suspended) Gold: Earned 500,000 credits (744,531)Generalized Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,007,004)PPS Sieve Double Gold: Earned 500,000,000 credits (570,848,729)TRP Sieve (suspended) Bronze: Earned 10,000 credits (21,181)AP 26/27 Sapphire: Earned 20,000,000 credits (20,845,708)WW Double Gold: Earned 500,000,000 credits (857,724,000)GFN Double Silver: Earned 200,000,000 credits (479,187,571)
Message 126855 - Posted: 14 Feb 2019 | 23:39:40 UTC
Last modified: 14 Feb 2019 | 23:58:04 UTC

I hope this isn't too far away from suitable topics.

Many years ago I played A LOT of Yahtzee and became interested in calculating the probability of getting a small straight. An assumption I made is that at each roll, the player keeps the dice that gives the highest probability of getting small straight. And another assumption is that the player is going for small straight from the start of their turn (and excluding yahtzee completely)

If you roll the dice, and don't have a 3 or 4, roll them all again. I have the following ways of getting a small straight having rolled before:

3,4 ---> 3ss (1234,2345,3456)..... P=468/1296
2,3,4 ---> 2ss (1234,2345)..... P=720/1296
1,2,4 ---> ss (1234)..... P=396/1296
2,4 ---> 2ss (1234,2345)..... P=324/1296
2,5 ---> ss (2345)..... P=180/1296

Probability of a small straight in 1 roll:
-------> ss (1234,2345,3456) P=200/1296

For the above examples, it was taken into account the following numbers are equivalent (and I am only looking at the numbers kept from the first roll):
2,3,4 = 3,4,5
1,2,4 = 1,2,3 = 3,5,6 = 4,5,6
2,4 = 3,5
2,5 = 1,5 = 1,6 = 2,6 = 1,2 = 5,6 = 1 = 2 = 5 = 6 (anything without a 3 or 4)

What I just wrote may be quite confusing, so I'll try to make it less confusing. For the last line, here are some (hopefully all) of the possible rolls that have two numbers and no 3s or 4s:
2,5,5,5,5 2,2,5,5,5 2,2,2,5,5 2,2,2,2,5
1,5,5,5,5 1,1,5,5,5 1,1,1,5,5 1,1,1,1,5
1,6,6,6,6 1,1,6,6,6 1,1,1,6,6 1,1,1,1,6
2,6,6,6,6 2,2,6,6,6 2,2,2,6,6 2,2,2,2,6
1,2,2,2,2 1,1,2,2,2 1,1,1,2,2 1,1,1,1,2
5,6,6,6,6 5,5,6,6,6 5,5,5,5,6 5,5,5,5,6
1,1,1,1,1 2,2,2,2,2 5,5,5,5,5 6,6,6,6,6

To calculate small straight I broke the first roll into different forms and went from there:
ABCDE..... P=120/1296
AABCD..... P=600/1296
AABBC.....P=300/1296
AAABC.....P=200/1296
AAABB.....P=50/1296
AAAAB.....P=25/1296
AAAAA.....P=1/1296

Some more boring stuff. I categorised types of rolls (so I could add the probabilities of these same types together). And the following is for having 4 different numbers of the form AABCD:
1234 - ss
123 5 - type3 ("double choice")
12 45 - type3 ("double choice")
1 345 - type1
2345 - ss
123 6 - type4
12 4 6 - type4
1 34 6 - type2
234 6 - type1
12 56 - type5 (roll them all again)
1 3 56 - type4
23 56 - type3 ("double choice")
1 456 - type4
2 456 - type3 ("double choice")
3456 - ss

What I mean by double choice is (and using 1235 as an example):
From 1235, you can either keep 123 or 235.

For a first roll, the probability of the form AABCD is P = 600/1296
For a 2nd roll, the following probabilities:
P(ss) = (3/15)*(600/1296) = 120/1296
P(type1) = (2/15)*(600/1296) = 80/1296
P(type2) = (1/15)*(600/1296) = 40/1296
P(type3) = (4/15)*(600/1296) = 160/1296
P(type4) = (4/15)*(600/1296) = 160/1296
P(type5) = (1/15)*(600/1296) = 40/1296

At some stage I may have played Yahtzee where you could score a yahtzee bonus as a straight. I think that is very not allowed. A yahtzee (5 of a kind) scored as a straight, which has to have at least 4 numbers different and in a row.

Anyway, the calculations I made for small straight rightly not included anything to do with yahtzee bonuses.

PROBABILITY OF SMALL STRAIGHT = 58.4899617%

(I need to double check this - I think I made a wrong assumption along the way)

My lucky highest score for Yahtzee is 710 with 4 or 5 Yahtzee bonuses (though I may have to take off 40 points as I strongly suspect I scored a high straight with a YB)

I can add a link to a folder in dropbox with images of the calculations, if that is ok to do. If someone could please advise me if that is ok, that would be great.

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Found 1 prime in the 2020 Tour de Primes321 LLR Gold: Earned 500,000 credits (529,293)Cullen LLR Gold: Earned 500,000 credits (611,298)ESP LLR Silver: Earned 100,000 credits (174,818)Generalized Cullen/Woodall LLR Bronze: Earned 10,000 credits (35,236)PPS LLR Jade: Earned 10,000,000 credits (12,687,171)PSP LLR Silver: Earned 100,000 credits (212,242)SoB LLR Silver: Earned 100,000 credits (466,812)SR5 LLR Silver: Earned 100,000 credits (145,419)SGS LLR Silver: Earned 100,000 credits (112,277)TRP LLR Silver: Earned 100,000 credits (342,501)Woodall LLR Silver: Earned 100,000 credits (109,455)321 Sieve (suspended) Silver: Earned 100,000 credits (175,037)PPS Sieve Bronze: Earned 10,000 credits (10,113)AP 26/27 Bronze: Earned 10,000 credits (12,129)WW Turquoise: Earned 5,000,000 credits (9,640,000)GFN Ruby: Earned 2,000,000 credits (2,059,029)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 126863 - Posted: 15 Feb 2019 | 7:08:01 UTC

You may want to ask at https://math.stackexchange.com or https://stats.stackexchange.com (where people earn points for providing good answers). /JeppeSN

Profile NickProject donor
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Joined: 11 Jul 11
Posts: 956
ID: 105020
Credit: 2,232,615,487
RAC: 7,116,244
Discovered 8 mega primesFound 5 primes in the 2020 Tour de PrimesFound 2 mega primes in the 2020 Tour de PrimesFound 4 primes in the 2021 Tour de PrimesFound 2 mega primes in the 2021 Tour de Primes321 LLR Sapphire: Earned 20,000,000 credits (20,020,701)Cullen LLR Sapphire: Earned 20,000,000 credits (20,174,451)ESP LLR Sapphire: Earned 20,000,000 credits (23,496,803)Generalized Cullen/Woodall LLR Sapphire: Earned 20,000,000 credits (20,197,125)PPS LLR Emerald: Earned 50,000,000 credits (61,145,268)PSP LLR Sapphire: Earned 20,000,000 credits (22,678,721)SoB LLR Sapphire: Earned 20,000,000 credits (26,978,901)SR5 LLR Sapphire: Earned 20,000,000 credits (20,183,763)SGS LLR Sapphire: Earned 20,000,000 credits (20,482,069)TRP LLR Sapphire: Earned 20,000,000 credits (20,307,419)Woodall LLR Sapphire: Earned 20,000,000 credits (22,371,625)321 Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,380,527)Cullen/Woodall Sieve (suspended) Gold: Earned 500,000 credits (744,531)Generalized Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,007,004)PPS Sieve Double Gold: Earned 500,000,000 credits (570,848,729)TRP Sieve (suspended) Bronze: Earned 10,000 credits (21,181)AP 26/27 Sapphire: Earned 20,000,000 credits (20,845,708)WW Double Gold: Earned 500,000,000 credits (857,724,000)GFN Double Silver: Earned 200,000,000 credits (479,187,571)
Message 126917 - Posted: 16 Feb 2019 | 15:11:58 UTC - in response to Message 126863.

Thank you for your reply. I'll chase it up with the links you provided, if anyone becomes interested - which I doubt.

Profile NickProject donor
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Joined: 11 Jul 11
Posts: 956
ID: 105020
Credit: 2,232,615,487
RAC: 7,116,244
Discovered 8 mega primesFound 5 primes in the 2020 Tour de PrimesFound 2 mega primes in the 2020 Tour de PrimesFound 4 primes in the 2021 Tour de PrimesFound 2 mega primes in the 2021 Tour de Primes321 LLR Sapphire: Earned 20,000,000 credits (20,020,701)Cullen LLR Sapphire: Earned 20,000,000 credits (20,174,451)ESP LLR Sapphire: Earned 20,000,000 credits (23,496,803)Generalized Cullen/Woodall LLR Sapphire: Earned 20,000,000 credits (20,197,125)PPS LLR Emerald: Earned 50,000,000 credits (61,145,268)PSP LLR Sapphire: Earned 20,000,000 credits (22,678,721)SoB LLR Sapphire: Earned 20,000,000 credits (26,978,901)SR5 LLR Sapphire: Earned 20,000,000 credits (20,183,763)SGS LLR Sapphire: Earned 20,000,000 credits (20,482,069)TRP LLR Sapphire: Earned 20,000,000 credits (20,307,419)Woodall LLR Sapphire: Earned 20,000,000 credits (22,371,625)321 Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,380,527)Cullen/Woodall Sieve (suspended) Gold: Earned 500,000 credits (744,531)Generalized Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,007,004)PPS Sieve Double Gold: Earned 500,000,000 credits (570,848,729)TRP Sieve (suspended) Bronze: Earned 10,000 credits (21,181)AP 26/27 Sapphire: Earned 20,000,000 credits (20,845,708)WW Double Gold: Earned 500,000,000 credits (857,724,000)GFN Double Silver: Earned 200,000,000 credits (479,187,571)
Message 126918 - Posted: 16 Feb 2019 | 15:12:20 UTC - in response to Message 126855.

I'd love to edit what I wrote - some glaring errors. I can't find how to. In my favour, the topic is so dull, that the errors may not be noticed.

Message boards : General discussion : Probability of small straight in yahtzee

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