EX 3.35 animal population

• Scenario:

  • Five individuals from an animal population near extinction have been:
  • Caught, tagged, and released.
  • Purpose: To mix into the population.

• After mixing:

  • A random sample of 10 animals is selected.

• Define:

  • $X$: the number of tagged animals in the second sample.

• Given:

  • Total number of animals in the region: 25.
  • Number of tagged animals: 5.

• Parameter values:

  • $n=10$: number of animals in the sample.
  • $M=5$: number of tagged animals in the population.
  • $N=25$: total number of animals in the population.

• The probability mass function (pmf) of $X$ (the number of tagged animals in the sample) is defined based on these parameters.

  • The pmf can be calculated using the hypergeometric distribution:
  • $P(X=k)=\frac{\left(\genfrac{}{}{0ex}{}{M}{k}\right)\left(\genfrac{}{}{0ex}{}{N-M}{n-k}\right)}{\left(\genfrac{}{}{0ex}{}{N}{n}\right)}$
    • Where:
      • $k$: number of tagged animals in the sample (possible values range from 0 to $\min (n,M)$).

$h\left(x;10,5,25\right)=\frac{\left(\begin{array}{l}5\\ x\end{array}\right)\left(\begin{array}{c}20\\ 10-x\end{array}\right)}{\left(\begin{array}{l}25\\ 10\end{array}\right)}$ where $x=0,1,2,3,4,5$

The probability that

• exactly two of the animals in the second sample are tagged is $P\left(X=2\right)=h\left(2;10,5,25\right)=\frac{\left(\begin{array}{l}5\\ 2\end{array}\right)\left(\begin{array}{c}20\\ 8\end{array}\right)}{\left(\begin{array}{l}25\\ 10\end{array}\right)}=.385$

The probability that at most two of the animals in the recapture sample are tagged is

$$\begin{array}{rl}P(X\le 2)& =P(X=0,1,\text{ or }2)\\ & =\sum _{x=0}^{2}h(x;10,5,25)\\ & =0.057+0.257+0.385\\ & =0.699\end{array}$$