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August 6th, 2011, 01:09 PM  #1 
Senior Member Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0  Tossing dice
We have three dice in a box. A round consists in shaking the box and then removing all the dice which faced up 1. The game ends once all dice have been removed. Find the probability that the game ends after exactly rounds. After cumbersome calculations and casework, I reached a recursion which seems hard to be converted into closed form. Obviously My question is, can this be generalized for dice? Or even for different probabilities, i.e. dice numbered from one to ten? And what if I want to know the expected value of the number of rounds of a game? 
August 6th, 2011, 01:34 PM  #2 
Newbie Joined: Aug 2011 Posts: 5 Thanks: 0  Re: Tossing dice
To do expected value, wouldn't you just roll the dice and then divide the totals by the amount of times? Because if you had .30 as an expected value and you wanted to know that it's .30 it would mean that you gain .30 profit for every time. If you had like 3.00 profit and you got 10 the answer would be .30. For the question, since they're dice, you would probably have to do an actual experiment. 
August 6th, 2011, 01:35 PM  #3 
Newbie Joined: Aug 2011 Posts: 5 Thanks: 0  Re: Tossing dice
And that's because you made an even value for every time. If you roll the dice it will never be even every time so you'd probably have to have numbers for every try right?

August 6th, 2011, 02:16 PM  #4  
Senior Member Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0  Re: Tossing dice Quote:
 
August 6th, 2011, 04:14 PM  #5  
Newbie Joined: Aug 2011 Posts: 5 Thanks: 0  Re: Tossing dice Quote:
What an attitude  
August 6th, 2011, 04:44 PM  #6  
Senior Member Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0  Re: Tossing dice Quote:
Sorry if I sounded a bit too rough.  
August 6th, 2011, 08:19 PM  #7  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Tossing dice Quote:
The probability that a given die will be removed on round n is . The probability that it's removed before round n is. So knowing those two probabilities you can find the probability of the game ending in n rounds as .  
August 7th, 2011, 05:54 AM  #8  
Senior Member Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0  Re: Tossing dice Quote:
EDIT: Amazingly, WA simplifies your summation to And this seems correct; surprisingly, the solution to the recursive relation in the original post is indeed . With respect to the expected value; WA gives for the case k = 3. How can it be generalized for any k? (Wolfram Alpha couldn't) http://www.wolframalpha.com/input/?i...n1%29%29^3%29 How can those infinite sums be computed by hand?  
August 7th, 2011, 10:30 AM  #9  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Tossing dice Quote:
Quote:
 
August 7th, 2011, 12:27 PM  #10  
Senior Member Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0  Re: Tossing dice Quote:
 

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