Which wheel you spin sets your payout

Question

A game show has two prize wheels. With probability $0.6$ you are sent to the Bronze wheel, whose payouts are $10, 20, 30$ each equally likely. With probability $0.4$ you go to the Gold wheel, whose payouts are $0, 50, 100$ each equally likely. You then spin the chosen wheel once.

What payout should you expect?

Accepted Answer

There are two stages: which wheel you are sent to, then the spin itself. The law of total expectation (tower rule) says to solve each wheel separately, then blend by how often you land on it: E[	ext{payout}] = E\big[\,E[	ext{payout} \mid 	ext{wheel}]\,\big]. Step 1, average each wheel on its own. - Bronze wheel: E[	ext{payout} \mid 	ext{Bronze}] = \frac{10 + 20 + 30}{3} = 20. - Gold wheel: E[	ext{payout} \mid 	ext{Gold}] = \frac{0 + 50 + 100}{3} = 50. Step 2, blend by the chance of each wheel. E[	ext{payout}] = 0.6 	imes 20 + 0.4 	imes 50 = 12 + 20 = 32. You expect a payout of 32. T…

Which wheel you spin sets your payout

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