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        riddles >> medium >> Dice rolls
        
(Message started by: Altamira_64 on May 31^st, 2013, 11:17pm)

Title: Dice rolls
Post by Altamira_64 on May 31^st, 2013, 11:17pm

What is the probability of rolling:
at least one 6 in 6 dice rolls?
at least two 6 in 12 rolls?
at least three 6 in 18 rolls?

Title: Re: Dice rolls
Post by pex on Jun 1^st, 2013, 12:46am

The probability of rolling at least n sixes in 6n rolls is [hide]1 - sum(k=0..n-1) (6n choose k) (1/6)^k (5/6)^(6n-k)[/hide]. So:
for n=1, [hide]1 - (5/6)^6[/hide] = 0.665
for n=2, [hide]1 - (5/6)^12 - 12*(1/6)*(5/6)^11[/hide] = 0.619
for n=3, [hide]1 - (5/6)^18 - 18*(1/6)*(5/6)^17 - (18*17/2)*(1/6)^2*(5/6)^16[/hide] = 0.597
As n goes to infinity, a Central Limit Theorem (http://en.wikipedia.org/wiki/Central_limit_theorem) can be applied to show that the probability approaches 0.5.