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Message boards : Aggie The Pew message board : Probability of small straight in Yahtzee

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Nick

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Message 132054 - Posted: 14 Aug 2019 | 22:50:57 UTC

I posted this before, but not on AtP message board:

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

PROBABILITY OF SMALL STRAIGHT = 58.4899617%

Only read on if you are extremely bored and don't mind your brain dribbling out your ears:

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 = 1,1 = 2,2 = 5,5 = 6,6 (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)

Cheers,
Nick

Gary Craig
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Message 132055 - Posted: 14 Aug 2019 | 23:10:13 UTC - in response to Message 132054.

I remember Yahtzee. Haven't played it in 30 years. In my teen years we had "strip yahtzee". No I will not post photos.

gary

Nick

Joined: 11 Jul 11
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Credit: 859,505,264
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Message 132056 - Posted: 14 Aug 2019 | 23:32:12 UTC - in response to Message 132055.

I remember Yahtzee. Haven't played it in 30 years. In my teen years we had "strip yahtzee". No I will not post photos.

gary

That sounds like it would have been entertaining!
At college in late teens/early 20s we would often play for beers - 4 ones would get you a beer. I think this skewed the results a bit as everyone was going for ones as a priority. I think a yahtzee of ones got a round for everyone. The games rapidly lost focus. :)
Cheers, Nick

Gary Craig
Volunteer tester

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Message 132057 - Posted: 15 Aug 2019 | 0:22:10 UTC - in response to Message 132056.

Reminds me of the game of "beer golf". For every stroke over par you had to take a drink. It degenerates into chaos very quickly.

g

Nick

Joined: 11 Jul 11
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Message 132058 - Posted: 15 Aug 2019 | 2:01:17 UTC - in response to Message 132057.

Sounds like another very entertaining game - with rapidly increasing handicap

Lord Of The Rats

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Message 132063 - Posted: 15 Aug 2019 | 20:38:09 UTC - in response to Message 132054.

Most interesting. I shall enjoy pondering on it. I think it confirms my suspicion that a small straight is easy to get for the large number of points it scores.

Lord Of The Rats

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Message 132064 - Posted: 15 Aug 2019 | 20:48:44 UTC - in response to Message 132057.

Reminds me of the game of "beer golf". For every stroke over par you had to take a drink. It degenerates into chaos very quickly.

g

Rat Hall contains as you may recall Erin's "blade" which is an oar won under circumstances by teams of 8 rowers and a cox racing in the style of the Oxford and Cambridge Boat Race. I'll bore with the story another day.

When I first met Erin I explained that I had studied at one of the UK's leading sporting universities which is true. I also explained that I was a regular member of the University's boat racing team which was also true. She was less impressed to find that my boat racing team was 8 blokes sitting on the ground as if they were rowing a boat. Each had a pint of beer in his hand. The chap at the front consumed his beer and put the glass on his head. Then the person behind started on his beer and when finished put his glass on his head and so on to the back of the line. The first team to complete all 8 won the heat.

It did tend to get a bit messy by the finals.

Don't try this at home; leave it to the trained athletes.

Nick

Joined: 11 Jul 11
Posts: 520
ID: 105020
Credit: 859,505,264
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Message 132066 - Posted: 15 Aug 2019 | 21:45:17 UTC - in response to Message 132064.

At Queens college, Melbourne University, we had similar what we possibly called boat races. I remember Queen's college first and second team beating the other colleges. We had a lot of talent in the beer drinking. I was proudly in the first team.

Cheers, Nick

Nick

Joined: 11 Jul 11
Posts: 520
ID: 105020
Credit: 859,505,264
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Message 132067 - Posted: 15 Aug 2019 | 21:55:07 UTC - in response to Message 132063.

Most interesting. I shall enjoy pondering on it. I think it confirms my suspicion that a small straight is easy to get for the large number of points it scores.

I should get my 'stuff' together and put together a dropbox link with images of the pages of calculations. On the other hand, it could be interesting to see if anyone else can come up with a probability and see if it agrees with what I found.

Rick Reynolds

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Message 132076 - Posted: 16 Aug 2019 | 14:01:38 UTC - in response to Message 132057.

Reminds me of the game of "beer golf". For every stroke over par you had to take a drink. It degenerates into chaos very quickly.

g

We drink regardless :) sometimes it makes our game better and sometimes...

Message boards : Aggie The Pew message board : Probability of small straight in Yahtzee