Expected value of seven bills from a box of eight

Question

A box holds eight banknotes: one $100 bill, three $20 bills and four $10 bills. You pull out $7$ of them at random without replacement. What is the expected total value of the bills you pull out?

Short answer

$175. The eight bills total $200, so they average $25. Each draw is marginally a uniformly random bill, so by linearity seven draws average $7\times25=175$ . Equivalently, the one bill left behind is worth $25 on average, and $200-25=175$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

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

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