Heads earn card draws until the first tail

Question

You flip a coin that lands heads with probability $\tfrac{2}{3}$ , again and again, stopping the instant it lands tails. For each head you got before that first tail, you draw a card worth a value equally likely to be $1, 2, 3, 4,$ or $5$ .

What total card value should you expect?

Accepted Answer

The number of cards you draw, call it N, is the number of heads before the first tail, which is random. Each card's value is random too. Condition on N and use the tower rule: E[	ext{total}] = E\big[\,E[	ext{total} \mid N]\,\big]. Step 1, freeze the count. If you know N = n, you draw n independent cards, each averaging \frac{1+2+3+4+5}{5} = 3. So E[	ext{total} \mid N = n] = 3\,n. Step 2, find the expected count. N counts heads before the first tail. Heads (probability 	frac23) are the "keep going" outcomes and a tail (probability 	frac13) stops you. The expected number of heads before th…

Heads earn card draws until the first tail

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