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. |