# crowing crows

##### Baruch MFE Faculty
20 crows land randomly on a wire. Each crow is crowing at the nearest crow. What is the expected number of crows that are not crowed at?

#### pruse

The end crows each have probability 1/2 of not being crowed at. The penultimate crows on each end have 0 probability of not being crowed at. For each "interior" crow, the probability of not being crowed at is 1/4. So the expected number of crows that are not crowed at is (2\cdot\frac{1}{2}+16\cdot\frac{1}{4}=5).

The end crows each have probability 1/2 of not being crowed at. The penultimate crows on each end have 0 probability of not being crowed at. For each "interior" crow, the probability of not being crowed at is 1/4. So the expected number of crows that are not crowed at is (2\cdot\frac{1}{2}+16\cdot\frac{1}{4}=5).

Maybe I misread the question, "each crow is crowing at the nearest crow" , doesn't this imply that once the first one lands every crow that follows must land adjacent to another crow? Thus only leaving the two edges crows not fully crowed.

This would give 2*1=2 , or is it me who misread it?

#### pruse

Maybe I misread the question, "each crow is crowing at the nearest crow" , doesn't this imply that once the first one lands every crow that follows must land adjacent to another crow? Thus only leaving the two edges crows not fully crowed.

This would give 2*1=2 , or is it me who misread it?

how is that? if 5 crows land on the number line at 0, 1, 3, 5, 6, then 3 isn't being crowed at, while the others are, including the end ones.

#### AlexandreB

C++:
% corroborated in matlab
clear; clc;
n=100000;
POS=sort(rand(20,n));
DIST=[repmat(realmax,2,n); POS(2:end,:)-POS(1:end-1,:); repmat(realmax,2,n)];
notCrowedAt=(DIST(1:end-3,:)<=DIST(2:end-2,:))&(DIST(3:end-1,:)>=DIST(4:end,:));
mean(sum(notCrowedAt)) % result ~ 5

#### lordvid

Maybe I misread the question, "each crow is crowing at the nearest crow" , doesn't this imply that once the first one lands every crow that follows must land adjacent to another crow? Thus only leaving the two edges crows not fully crowed.

This would give 2*1=2 , or is it me who misread it?

I think you are supposing that the crows are landing one at a time on the wire. Even so, if a new crow lands between 2 old crows, it will choose the nearer old crow to crow at, and the other old crow will not be crowed at. The new crow need not necessarily land at the ends of the wire.....

This puzzle needs songs by Counting Crows to accompany it....

how is that? if 5 crows land on the number line at 0, 1, 3, 5, 6, then 3 isn't being crowed at, while the others are, including the end ones.

I agree that what I understood makes a trivial problem, but to me the wording is a bit problematic.
I understood it as nearest=adjacent which means 0,1,2,.... .
I agree that how you understand it is the correct way and solution.

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