Non-red gumballs score points until a red one

Question

A gumball machine gives a red gumball with probability $\tfrac14$ each time, independently. You keep buying gumballs until you get a red one, then stop. For each non-red gumball you got before the red one, you flip a fair coin and score $0$ points on tails or $8$ points on heads.

What total score should you expect?

Accepted Answer

Let N be the number of non-red gumballs before the first red one, a random count, and each of those scores a random 0 or 8. Condition on N and use the tower rule: E[	ext{score}] = E\big[\,E[	ext{score} \mid N]\,\big]. Step 1, freeze the count. Each non-red gumball scores 0 or 8 with equal chance, averaging \frac{0 + 8}{2} = 4. If N = n, E[	ext{score} \mid N = n] = 4\,n. Step 2, find the expected count. A red gumball stops you (probability 	frac14), a non-red lets you continue (probability 	frac34). The expected number of non-red gumballs before the first red is E[N] = \frac{P(	ext{non-r…

Non-red gumballs score points until a red one

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