A random coin from a box, then one flip

Question

A box holds three coins: two ordinary fair coins and one trick coin that has heads on both sides (so it always lands heads). You pick one coin at random and flip it once, scoring $1$ for heads and $0$ for tails.

What is your expected score?

Accepted Answer

There are two branches depending on which coin you grabbed. The law of total expectation (tower rule) says to average each branch's result, weighted by how likely that branch is: E[	ext{score}] = E\big[\,E[	ext{score} \mid 	ext{coin}]\,\big]. Step 1, score each branch. - Fair coin: heads with probability 	frac12, so E[	ext{score} \mid 	ext{fair}] = 0.5. - Trick coin: always heads, so E[	ext{score} \mid 	ext{trick}] = 1. Step 2, weight by how often each coin is picked. Two of the three coins are fair and one is the trick coin, so P(	ext{fair}) = 	frac23 and P(	ext{trick}) = 	frac13…

A random coin from a box, then one flip

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