If you were a scientist studying dolphins how might you try to estimate how many dolphins were in an ocean?

One way of doing this is to count the number of dolphins in a certain area and then take this as the population density for the whole ocean, eg. if there were 5 dolphins in 10 $km^{3}$ you could use this to estimate that there would be 500 dolphins in 1000 $km^{3}$
of water. Can you think of any problems with this approach? What assumptions does it make? Do you think that they are valid?

Another possible method is to capture dolphins over a period of time, tag them and release them. You can then catch another set of dolphins and see how many of them are tagged. For example, you could catch 100 dolphins from the wild and tag them before releasing them where they were caught. If you then catch another 100 dolphins and find that 5 of them had been tagged you could estimate that
because $\frac{5}{100}$ dolphins in your sample were tagged, 5% of all dolphins had been tagged. You tagged 100 dolphins so if this were true there would be $\frac{100}{0.05}$ = 2000 dolphins in total. What assumptions does this method rely on?

Which of the two methods do you think is better? Is one always better than the other or are there some conditions (eg size of ocean, size of total dolphin population) that might favour the other better? Can you think of any ways to improve these approaches?