Question

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?

Answer 1

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.