German Tank Problem

Relevant wiki: Maximum Likelihood Estimation (MLE)

If $N < 17,$ the probability is, of course, 0. If $N\ge 17,$ the probability of seeing these three planes is $3! \times \frac{1}{N} \times \frac{1}{N-1} \times \frac{1}{N-2}.$ Note that as $N$ grows, this probability decreases, so the maximum likelihood estimate for $N$ is 17.

This should strike you as slightly weird -- intuitively, 17 seems like a bad guess for $N$ since, well, "what are the odds I happened to see the largest plane?" This illustrates an issue with maximum likelihood estimators (finding the parameter which makes the observed outcome most likely). A better estimate might be the minimum-variance unbiased estimator*, which is $m \cdot \left(1+\frac{1}{k}\right)-1 = 17 \cdot \left(1+ \frac{1}{3}\right) - 1 \approx 21.67.$

*: Unbiased estimator means that the expected value of the estimate for the parameter is equal to the true value of the parameter. The minimum-variance one is often considered the "best", which should be fairly intuitive since lower variance means your estimate is more likely to be close to the true value. In general, the bias-variance tradeoff is a fundamental issues in fields like statistics and machine learning .

I chose the lowest number possible that included one of the selected values because then the probability of the given numbers being selected is higher compared to the other N values

The numerator is fixed, and the bigger the N is, the bigger the denominator is. Then we choose min N 17