In a lake there are 53 blue tail fishes and 40 yellow tail fishes. we capture 12 fishes. what is the most probable number of blue tail fishes?

In general, the most probable distribution is in the ratio of the respective types, namely

Number of blue tails

=12 * (53 / (53+40))

=6.84

To check between 6 or 7, we use the hypergeometric distribution:

P (B=6)

=C (40,6) C (53,6) / (C (93,12)

=3838380*22957480/416579843773639

=0.211530

P (B=7)

=C (40,5) C (53,7) / (C (93,12)

=658008*154143080/416579843773639

=65563917120/269282381237

=0.243476

Hence the most probable number of blue tail fishes is 7

Note: it would be prudent to calculate P (B=8) = 0.194443 to show that P (B=7) is indeed the highest value.