Money tripled or halved by a coin, three rounds

Question

You start with \10. Each round you flip a fair coin: heads triples your money, tails halves it. You play three rounds. What amount should you expect to hold at the end?

Accepted Answer

Each round multiplies your money by a random factor, and the rounds are independent. Condition on the money before each round with the tower rule. Step 1, the average multiplier per round. The factor is 3 on heads or 	frac12 on tails, each with probability 	frac12, so E[	ext{factor}] = 	frac12 	imes 3 + 	frac12 	imes 	frac12 = 1.5 + 0.25 = 1.75. Step 2, the recursion. Let Mt be your money after t rounds. Given M{t-1}, the next round multiplies it by an independent factor averaging 1.75, so E[Mt \mid M{t-1}] = 1.75 \, M{t-1}, \qquad 	ext{hence} \qquad E[Mt] = 1.75 \, E[M{t-1}]. Step 3…

Money tripled or halved by a coin, three rounds

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