Expected birthday pairs

What this expected-value probability question tests

This is an easy probability question that probes your ability to apply linearity of expectation to count pairs. It is a natural variant of the classic birthday-paradox problem, and quant firms use it to check whether you can set up the right framework quickly, even when the setup differs slightly from the canonical version.

The key insight is to define indicator random variables for each possible pair, compute the probability that any two people share a birthday, and then sum across all pairs. This approach bypasses the need to enumerate overlapping events and scales cleanly. Interviewers value clean problem setup and correct use of linearity of expectation over numerical speed.

• Linearity of expectation and indicator variables
• Pairwise independence and counting
• Contrast with the birthday paradox (probability of at least one match)