The Last Record Low

What this record-low expected-value question tests

This is a medium-difficulty probability question that asks you to find the expected number of record lows in a shuffled deck. It is a classic test of whether you can apply linearity of expectation to decompose a counting problem, rather than trying to enumerate outcomes directly.

The key insight is recognizing that you can express the total count as a sum of indicator random variables—one for each position—and then compute the probability that each position holds a record low. This approach avoids the combinatorial explosion that comes from tracking the full joint distribution of records.

• Linearity of expectation and indicator decomposition
• Computing marginal probabilities under uniform random permutation
• Recognizing symmetries in random orderings