Heads in five tosses when the ends disagree

Question

A fair coin is tossed five times. Given that the first and fifth tosses show different faces, what is the expected total number of heads?

Short answer

$\tfrac52$ . The two end tosses contribute exactly one head between them, and the middle three tosses are unaffected by the condition, contributing $\tfrac32$ on average.

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Lessons behind this question

Conditional Expectation and Variance Revisited
Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating

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