Expected value of a die times a coin

Question

You roll one fair die and flip a fair coin that is worth $1$ on heads and $0$ on tails. You are paid the die times the coin, in dollars. Solve in your head.

What is the fair price of this game? Aim: under 10 seconds.

Accepted Answer

The die and coin are independent, so the product's expectation factorises: E[XY] = E[X]\,E[Y] = 3.5 	imes 0.5 = 1.75. The die averages 3.5; the coin averages \frac{0 + 1}{2} = 0.5. Half of 3.5 is 1.75. Intuition check: half the time (tails) you are paid 0; the other half you are paid the die's average 3.5. So 	frac12 	imes 0 + 	frac12 	imes 3.5 = 1.75, the same answer from the other direction. Takeaway. E[XY] = E[X]E[Y] under independence, even when the two variables live on totally different scales. A 0-or-1 multiplier just halves the other factor's mean.

Expected value of a die times a coin

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