Hi, i some time ago my physics teacher asked me this problem: There are 20 chickens coming out of an hen-coop. 15 are white, 5 are black. What's the probability of at least 2 black chickens come out consecutively.(adsbygoogle = window.adsbygoogle || []).push({});

so basicly i tried to calc P = 1 - (Probability of no black chicken come out consecutively )

So i thought P = 1 - D(20,5)/C(20,5)

being D = Ways to arrange the 5 chickens in 20 "boxes" so that the 5 black never come out consecutivelly

after some thinking i figured out that

D(n,p) = D(n-2,p-1) + D(n-1,p)

which is somewhat similar to C(n,p) = C(n-1,p-1) + C(n-1,p)

So i started making some sort of pascal triangle for the D function which lead me to this complicated sum:

[tex] D(n,p) = \sum_{i=1}^{n-2p+2} C(p-3+i,p-2) \times (k-i+1) [/tex]

hope i got this right (first time using latex ^^)

well, i came to the same result as my teacher but he had a much more simple formula, to which he arrived empirically:

P = 1 - C(20-5+1,5)*15!*5!/20!

So, D(n,p)=C(n-p+1,p) i tried to deduce this formula from what i previously had but with no success. Can some1 plz tell me how i could have arrived to it.

Ty

PS: By the way, i'm only in highschool so try to keep it simple.

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# 20 chickens

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