Thread 16763264

There are seven ducks in a circular pond in random locations. The diameter of the pond is 100 meters.

You decide to throw a piece of bread for the ducks to eat. The bread will land in a random location into the pond. But if it lands more than 10 meters away from a duck, the duck won't hear the splash and it won't notice the bread. Otherwise it will come and eat it.

What is the probability that the bread will get eaten once you throw it?

Pond area = pi * 100^2m
Chance to meat a duck in 1^2m = 10 / Pond area
Bread area = pi * 10^2m
Average number of ducks in bread area = 10 * Bread area / Pond area = 10 * 10^2 / 100^2 = 1000 / 10000 = 0.1

This number is less then one and can be interpreted as probability: 10% chance.

But deep inside I feel that in simulated experiment this number will be different. It doesn't consider edge case when the bread lands near the shore and it's area getting cut. So I'll try to fix it by increasing pond radius by bread radius: R = 110

In this case:
p = 10 * 10^2 / 110^2 = 10 / 11 * 11 = 0.083

>>16763283
I still feel that answer is not correct, because with the increased radius I didn't consider that duck still are enclosed in pond radius 100.

58%

About 130000 ducks make a monolayer that guarantees you hit a duck with the bread, which won't make a splash to attract the ducks.

>>16763264 (OP)
one duck's hearing covers 1/100th of the area of the pond, which means all of them cover at most 10% and at the least a tiny bit over 0.1%
more precision demands picking some behavior for how the ducks are distributed or behaving on the pond, which probably ends up being clustered somehow so I'd vaguely gesture at 3% as my final answer because birds of a feather that quack like ducks tend to flock together like ducks

>>16763523
Nice theory, but in reality there will be some deviation from the ideal monolayer and a clusterduck will form.

When the bread lands near the edge, the "target" area (for ducks) is cut off by the edge of the pond. So you need to integrate an expression for the overlap of two circles (pic related) between a couple of values. Looks impossible but there might be a way since the terminals will be limiting cases. Specifically you'd have to integrate something like pic related from d = R - r to d = R, or similar.

>>16764262
According to wolfram, these integrals are actually doable (when multiplied by r aka "x" for the area element), so there should be a nice-ish solution. There might be a better way to get there than doing these integrals thoughbeit.

>>16763264 (OP)
100%

Pond radius(R) = 50m
Pond area = pi * 50^2 sq m

One Bread splash radius(r) = 10m
One Bread splash area(a) = pi * 10^2 sq m

Total Bread splash radius not considering pond (R +r) = 60m
Total Bread splash area(A) = pi * 60^2 sq m

Chance of one duck to eat bread (p1) = One Bread splash area / Pond area = 1/25

This is incorrect as Bread that lands on edge of pond, has a Bread splash area in shape of cresent of moon
Hence a larger area with Total Bread splash radius has to be taken.

Corrected chance of one duck to eat bread (p) = One Bread splash area / Total Bread splash area = a/A = 1/36

The problem reduces to, if one duck has p probability of eating bread, out of seven ducks what is probability at least one eats bread.

Probability none of duck eats bread = (1-p)^7
Probability at least one gets to bread = 1 -(1-p)^7 = 0.99