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On a long, slow trip to Mars, Virgin Galactic is entertaining it's passengers with a prize drawing.

Every hour there is a 40% chance you'll win. If you don't win, then you have the same odds to try again and again each hour. Prizes follow this sequence: $1, 1, 2, 2, 5, 5, 100, 100.

The sequence restarts after not winning 8 straight or if the passenger wins before then. The ship's computer automatically enters you in the contest. Passengers need not be present or awake to play, nor is there a cost to play.

What is Virgin's expected cost per hour, per passenger? (Or, what is the expected value of each hour of your travel?)

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The expected winnings for each sequence restart is the sum of probailities of getting each prize * values of the prize, or $0.4\times 1 + 0.6\times 0.4\times 1 + 0.6^2 \times 0.4 \times 2+... + 0.6^7\times 0.4\times 100=4.501504$. The expected number of hours before the sequence restarts is $0.4\times 1+0.4*0.6\times 2+...+0.4*0.6**7*8=2.32364032$. So expected value per hour of travel is $\frac{4.501504}{2.32364032}=\$2.5$

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