Student members of IMS are invited to submit solutions to bulletin@imstat.org (with subject “Student Puzzle Corner”). The names of student members who submit correct solutions, and the answer, will be published in the issue following the deadline. The Puzzle Editor is Anirban DasGupta. His decision is final.

Student Puzzle Editor Anirban DasGupta poses a probability problem. The problem is a relatively simple one in the domain of asymptotic probability. The requested probability would be difficult to calculate exactly, and yet it would be possible to get an accurate approximation. Here is the exact problem:

250 people are each asked to toss a fair coin 10,000 times. Give an approximation with justification to the probability that at least four of them would have obtained exactly 5,000 heads and 5,000 tails.

Simulation-based answers will not be accepted.

### Solution to Puzzle 38

A reminder of the puzzle is here. Anirban explains:

We assume as our model that the observed paw prints are uniformly distributed in the circle centered at $(x_0, y_0)$ and with radius $\rho$. We may assume without loss of any generality that the center is the origin $(0, 0)$. Then denoting the locations of the paw prints as $\bf{X_i}$, the MLE $\hat{\rho}$ of $\rho$ is $\max(||\bf{X_i}||)$.
Its exact CDF is given by $P(\hat{\rho} \leq \rho \,x) = x^{2n}, 0 < x < 1$. Consequently, $\frac{n(\rho - \hat{\rho})}{\rho }$ converges in law to an exponential with mean $\frac{1}{2}$.