Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 18

Write $p_j=\mathbb{P}(X=j)$, with ${\sum }_{j=0}^N{p}_j=1$. The expected value, variance, and strict lower-tail probability areThe upper summation limit $Y-1$ is essential because the inequality is strict and the variable is integer-valued.

Count unordered sets of distinct labelled fish: there are $\left(\genfrac{}{}{0ex}{}{N}{M}\right)$ possible hauls. To obtain exactly $j$ trout, choose $j$ labels from the $K$ trout and $M-j$ from the $N-K$ carp, giving $\left(\genfrac{}{}{0ex}{}{K}{j}\right)\left(\genfrac{}{}{0ex}{}{N-K}{M-j}\right)$ possibilities. Uniform sampling without replacement therefore gives the hypergeometric distributionOutside this integer range the probability is zero. If ordered catches were counted instead, every haul would have $M!$ orders and that factor would cancel from the probability ratio.

Here equal catchability means equal selection probability for each individual remaining fish. It does not mean that the two species have probability $1/2$ on each draw regardless of their populations. This interpretation is the one that yields the stated hypergeometric law.

The total number of two-fish hauls is $\left(\genfrac{}{}{0ex}{}{10}{2}\right)=45$. Selecting both from the eight carp gives

Choose one of the two trout and one of the eight carp. The hypergeometric distribution gives

Only one labelled haul contains both trout, giving

The three preceding probabilities satisfyTheir expected value and second moment areSubtracting the squared mean givesThe value $\mu =2(2/10)$ also follows from linearity of expectation. The variance includes the finite-population reduction caused by sampling without replacement.

When only a tiny fraction of either species is removed, the trout fraction among the remaining fish stays nearly constant at $p=K/N$. Successive catches are then approximately independent trials with that fixed success probability. The count therefore approaches the binomial distributionThe approximation comes from negligible depletion, not from an exact independence property of sampling without replacement.