For those who don't understand the question, here are the rules:

1. Five six-sided dice are rolled.

2. The object is to have all of them on the same face, for example all twos.

3. After any roll you may hold any dice you wish and re-roll the others.

4. Switching to a different number is allowed. For example if you roll 1-1-2-3-4 on the first roll, then hold the ones, and then roll a 2-2-2 with the other three, having 1-1-2-2-2, then you can switch to holding the twos and re-roll the ones.

Normally in the game of Yahtzee you get only three rolls, but I'm allowing ten (to make it harder).

For extra credit, what is the probability after 1 to 9 rolls?

Quote:WizardWhat is the probability of achieving a Yahtzee after ten rolls?

For those who don't understand the question, here are the rules:

1. Five six-sided dice are rolled.

2. The object is to have all of them on the same face, for example all twos.

3. After any roll you may hold any dice you wish and re-roll the others.

4. Switching to a different number is allowed. For example if you roll 1-1-2-3-4 on the first roll, then hold the ones, and then roll a 2-2-2 with the other three, having 1-1-2-2-2, then you can switch to holding the twos and re-roll the ones.

Normally in the game of Yahtzee you get only three rolls, but I'm allowing ten (to make it harder).

For extra credit, what is the probability after 1 to 9 rolls?

So, you're allowing UP TO 10 rolls, stopping on any Yahtzee, in the original question, right? Which means the extra credit question would also be all-inclusive (additive)? Or you're looking for EXACTLY each number of rolls, not less (I don't think this is the question, but making sure).

..after that it is pretty straightforward to calculate the probability.

If you get a Yahtzee before the end, just keep all five dice. So both questions are equivalent.Quote:beachbumbabsSo, you're allowing UP TO 10 rolls, stopping on any Yahtzee, in the original question, right? Which means the extra credit question would also be all-inclusive (additive)? Or you're looking for EXACTLY each number of rolls, not less (I don't think this is the question, but making sure).

I love Yahtzee,Quote:WizardWhat is the probability of achieving a Yahtzee after ten rolls?

well used to play it a lot and changing rules to the game.

This actually in school was the first exercise in combinations and matrix algebra.

I could not do this in class. I had to have lots of help.

since then I have some of this in Excel for the 5 kinds

and the matrix raised to 10

so I think you want as an answer 1-[1,6]

even show the expected number of visits to each state (yellows)

I also have this just using recursion in Excel. many do not know how to raise a matrix to a power in Excel and that is OK.

a few formulas in a few cells is all that is required for the recursion method

I will link to it as soon as I finish it.

looks like this for the first 50 rolls

here is one place to start for any interested in a quick read, there really are many of them

http://www.datagenetics.com/blog/january42012/

Sally

Quote:kubikulannIf you get a Yahtzee before the end, just keep all five dice. So both questions are equivalent.

Yes, keeping all five is allowed? So asking about a Yahtzee in ten rolls could also be stated as "ten or less."

Quote:chrisrI came up with it being slightly more advantageous to re-roll all 5 dice if you don't have at least a pair.. The numbers i came up with for Yahtzee in 10 or less rolls were 0.545 and 0.550 depending on if you keep an arbitrary single die when you have less than a pair.

edit: That is what I get for trying to be clever and then my spoiler tags don't work either :)

Quote:chrisrI came up with it being slightly more advantageous to re-roll all 5 dice if you don't have at least a pair.. The numbers i came up with for Yahtzee in 10 or less rolls were 0.545 and 0.550 depending on if you keep an arbitrary single die when you have less than a pair.

This has to be a rounding error. Keeping all 5 and keeping 4 out of 5 are identical.

To see this, imagine that, once you decide which dice to re-roll, you roll them one at a time. If you re-roll all 5, doesn't matter how the first die comes up.