This is part of a free course in Undergraduate Probability: https://www.math.uci.edu/~rvershyn/teaching/ugp/ugp.html
0:00 We state the secretary problem,
1:16 and propose a stopping strategy to solve it
2:00 using a conditioning trick.
2:24 Let's condition on the position of the best prize
2:42 and use the law of total probability.
3:35 If the best prize is among the first k, we never win it.
3:55 If it is not, the probability to win it is k/(i-1).
6:09 Plug this into the law of total probability to get the answer for each k.
7:23 The answer can be approximated using integration,
8:42 optimized in k using differentiation,
9:59 which gives the ultimate answer to the secretary problem.
10:28 The probability to win the best prize is 37% even if the number of prizes is large.
