From Francesca at Wimbledon High School : by the end of that week, the $40$ marked fish would have spread out and integrated with the rest of the fish population. So that the second lot of $40$ fish that are caught would be a thoroughly mixed group of marked and unmarked fish.
For example if in the second lot of $40$ fish, the marked to unmarked ratio was $2 : 38$ and we know that there are $40$ marked fish altogether, we might assume that the ratio in the second lot of fish is close to the ratio for all the fish together.
Hey presto!!! There are $800$ fish altogether!
For example if when you looked at the second catch and $10$ were already marked then you could estimate that you had marked a quarter of the entire population of fish, because that's the proportion in the sample, so the original $40$ were about one quarter of the whole population and there are therefore approximately $120$ fish.
Mathematical reasoning & proof. Binomial distribution. Handling data. Interpreting data. Sampling and hypothesis testing. Fractions. Processing and representing data. Collecting data. Sampling. Biology.