I'm having trouble understanding this famous problem known as the German tank problem. The problem goes as follows:
Assuming tanks are assigned sequential serial numbers starting with 1, suppose that four tanks are captured and that they have the serial numbers: 19, 40, 42 and 60. What's your best guess for $N$, the total number of tanks that are produced?
The Wikipedia article attached goes into more detail about how the frequentist and Bayesian POVs give you two slightly different answers, 74 and 89 respectively.
What I'm confused by is why the answer is not 60 for either of these methods? The 4 data points shown above (19, 40, 42, 60) seem most likely to happen if there are only 60 tanks. The more tanks there are, the less likely these data points are, simply just because there are more tanks. What am I missing?
The comment by @lulu helped a lot; I had found the MLE: the mode, not the average. While out of all possible choices for $N$, 60 is the most probable out of them, it's a biased choice to estimate $N$. It'll always be less than the median or mean choice for $N$.
By adding a uniform prior on the distribution of $N$s and using the likelihood function, you can derive the Bayesian answer that the article gives. Here's more detail on that.
And here's a StackExchange post on why the sample max is the only thing needed for analysis. I think it provides some clarity.