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Scientists randomly caught 30 fruit bats at an orchard. They tagged the fruit 
bats and then released them. Several weeks later, they captured 20 fruit bats at the same location. They found that 4 of those fruit bats 
had tags. Assume the population of fruit bats does not change. About how many fruit bats
 are at that orchard?

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  1. There are two ways to solve this problem.

    A.
    The intuitive (and fast) approximation is by prorating the ratio, assuming that the ratio is constant.
    Let N=total population, then
    N/30=20/4 => N=150

    B.
    The more accurate (and elaborate) way is to use the hypergeometric distribution. This distribution is typically used in instances where sampling is done without replacement, such as drawing cards out of a deck.
    Let
    A=recapture size = 20
    B=uncaptured size = population -20
    m=number of marked in sample (4)
    n=number of marked in the wild (30-4=26)
    then
    P(X=m)=C(A,m)*C(B,n)/C(A+B,m+n)
    C(n,r)=combination n choose r = n!/(r!(n-r)!)

    We need to find A+B.
    Substituting formula,
    P(x=4)=C(20,4)*C(B,26)/C(B+20,30)
    The value of P(x=4) is not directly known, but the maximum likelihood is when it is at the maximum for the various values of B.

    We will find that P(x=4) is at a maximum when B=129.5, which makes the total population A+B=20+129.5=149.5, quite close to our initial estimate.

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