Street poker.

At school my friends and me liked to play our own games with dices.
One of the games we invented was to throw with 6 cubical dices, each numbered from 1 to 6.
You had to throw 6 different pokers (6 ones, 6 twos etc.) and a street. You were free to choose or change the sequence after any throw.
We never agreed about the best strategy.
How many throws does one need (average), assuming one follows the best strategy?

Fjord

Originally posted by fjord
At school my friends and me liked to play our own games with dices.
One of the games we invented was to throw with 6 cubical dices, each numbered from 1 to 6.
You had to throw 6 different pokers (6 ones, 6 twos etc.) and a street. You were free to choose or change the sequence after any throw.
We never agreed about the best strategy.
How many throws does one need (average), assuming one follows the best strategy?

Fjord

Originally posted by iamatiger
To get this clear - Is this how it works?:

a)You throw all 6 dice (this counts as 1 throw).
b)You then note any poker or street that has appeared.
c)You then rethrow as many of the dice as you want (this is your second throw), and you carry on repeating b and c until you have got all the special combinations.

Originally posted by fjord
That is the way we played it.
Mind you, you are also allowed to pick up dices you left on the table during earlier throws.

E.g.
First throw: 1, 1, 2, 3, 4, 4.
You leave on the table: 1, 1,
Second throw: (1, 1) 3, 3, 3, 6
Now you may leave on the table: 3, 3, 3.

Or another variation for the second throw: 1, 1, 2, 3, 4, 5,
Now you may choose to pick up just one 1 and try to make a street.

Fjord

Originally posted by iamatiger
This is going to take some heavy analysis just to work out the best strategy. And maybe a computer program to work out the expected number of throws. Please be patient!

Here's the first part of the analysis:
If you throw 6 dice, the possible results are:

out of 46656 possible throws (in order of probability):

two pairs............16200 = 4 to a ...[text shortened]... you keep a partial straight (6 different)?
When should you keep a partial flush (6 of a kind)?

Originally posted by fjord
I will be patient ;-)
Just one question for the moment.
The probabilities you mention are right as long as you assume you will throw every time with six dice (sorry for the wrong plural I used in the previous posts; at school I learnt that ...[text shortened]... umber of probabilities you found. But maybe I'm wrong.

Fjord

Originally posted by iamatiger
Yes, one of the the next things I will calculate is your "probablity of improving" from various configurations of kept dice - but it is complex because it depends on what you have already got - for instance if you already got your 6 twos, the twos in your above example would be no use to you. But before I can this I think that I need to decide, if you should keep 1,1 or 1,2,3,4,5 from a dice throw of 1,1,2,3,4,5.

Originally posted by fjord
Let me give it a try.

According to your data the ratio of probabilities between throwing a flush of ones and a straight is 1 : 720. (1)
When you have got 1,1 on the table the probability of a flush of ones will be (after 46656 throws): 36
When you have got 1,2,3,4,5 on the table the probability of a straight will be (after 46656 throws): 7776.
So the rat ...[text shortened]... to 216.
This means we should go for the flush of ones after a throw of 1,1,2,3,4,5.

Fjord