Fish in a Pond

There are 5 fish in a pond. What is the probability that you can split the pond into 2 halves using a diameter, so that all fish end up in one half?

Let us generalize the problem to N fish in a pond. We can assume that all fish are on the boundary of the pond, which is a circle, and we need to find the probability that all of them are contained within a semi-circle.

For every fish Fᵢ, consider the semi-circle Cᵢ whose left end-point is at Fᵢ. The probability that all fish belong to Cᵢ is equal to 1/2ᴺ⁻¹. Since it is impossible to have 2 fish Fᵢ and Fⱼ, such that the semi-sircles Cᵢ and Cⱼ contain all fish, we see that the probability that all fish belong to Cᵢ for some i is equal to N/2ᴺ⁻¹.

When N = 5, we get that the answer is 5/16.

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Responses

  1. This does not consider the size of fish neither relative to each other, nor relative to the diameter of the pond. This problem seems to be assuming that all fish are dots. If so, it needs to be stated.

  2. Also you can think of the question of asking, what are the odds that if you randomly place 5 fish in circle, that they all end up on the desired side.
    Then you could use the binomial probability formula P(X=x) = C(n,x)*p^x*q^(n-x), where:
    X = # of occurrences of desired outcome: a fish being on the desired side
    n = total number of fish (5)
    x = number of fish we want to be on the correct side (5)
    p = probability of X occurring (0.5 because it has only 2 options)
    q = probability of X not occurring (0.5 for same reason as above)

    1. Hi Alexander, thank you for your comment. That’s right, you can use this formula, but only once you fix the semi-circle. Then, you will get C(4,4)(0.5)^4=1/16, which is what is implicitly done in the provided solution:) The main trick is to realize that:

      1. at most one of the semi-circles having as a left end-point a fish can contain all the other fish
      2. if all fish are contained within a semi-circle, then this semi-circle can be chosen such that it has a fish as a left end-point