Predict the Population of Puffins

Date: 09/28/2002 at 09:42:05
From: John Dyson
Subject: Predicting Population by Tagging and Spotting

Scientists were studying the population of puffins in Alaska. They 
spotted and tagged 52 puffins in December. Two months later, they 
spotted 50 puffins. Of those 50 puffins, 3/10 of them had been tagged 
in December. Predict the population of puffins in Alaska.

   A. 52 puffins
   B. 100 puffins
   C. 173 puffins
   D. 200 puffins

Date: 09/28/2002 at 16:46:31
From: Doctor Anthony
Subject: Re: Predicting Population by Tagging and Spotting

Suppose there are N puffins in Alaska. You tag 52 of them, so there 
are now 52 puffins with tags and N-52 without tags.

You then return and catch 50 more puffins. 15 are found to have tags 
and 35 without tags.

The probability of this event is  

            C(52,15) x C(N-52,35)
           -----------------------
                  C(N,50)

We choose the value of N which makes this probability a maximum.

We require the probability with N puffins to be greater than the 
probability with N-1 puffins.  So this gives

     C(52,15) x C(N-52,35)       C(52,15) x C(N-53,35)
     ------------------------  >  ---------------------
            C(N,50)                    C(N-1,50)


cancelling C(52,15) from each side this reduces to

        (N-52)!/[35!(N-87)!]        (N-53)!/[35!(N-88)!]
       ------------------------  >  ------------------------
          N!/[50!(N-50)!]           (N-1)!/[50!(N-51)!]

cancelling as much as we can from one side with the corresponding term 
on the other side we get:

        (N-52)/(N-87)
      ----------------  > 1
          N/(N-50)

         (N-52)/(N-87)  >  N/(N-50)

           (N-52)(N-50) > N(N-87)

      N^2 - 102N + 2600 > N^2 - 87N

                   2600 > 15N

and so   N < 2600/15 = 173.33

Similarly, comparing N with N+1 puffins in Alaska you will find that 
probabilities increase while N < 174 and then decrease again, so that 
the optimum value is 173.

So the probability of getting the second catch that they did is a 
maximum if N is 173.  This is answer C in the options available.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 09/28/2002 at 18:42:38
From: John Dyson
Subject: Thank you (Predicting Population by Tagging and Spotting)

I am most appreciative for your prompt attention to my problem. This 
was a question on a quiz for my Grade 8 math students and although I 
guessed the correct answer, I was unable to offer an explanation. Many 
thanks!