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Wednesday, August 8, 2018

Counting Chickens After They Hatch

Given n eggs, let the probability of an egg hatching be p, each hatching event being independent of the others. The probability of a chick surviving after hatching we'll call r, which is independent of the survival of other hatchlings. Let H be the number of eggs that hatch and C be the number of chicks that survive. What are the distributions of H and C?

Solution: H ~ Bin(n, p), which is pretty straightforward. We're given that post-hatch survival is C, with a probability independent of other hatched chicks. Chickens that end up surviving, according to this set up, go through two stages, hatching and surviving. We're also given that the probability of surviving after hatching is r, which is independent of p, the probability of hatching. So, the probability of survival for a single chick is $p * r$. This gives the distribution of C.

$H\sim Bin(n, p)$
$C\sim Bin(n, rp)$




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