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Problem: The jackpot for the California lottery is $100 million today. If I buy 1 ticket, what's my expected value? (only taking into account the jackpot prize, and not other possible winnings)

Solution:

    Based on the previous solution, the probability of winning with 1 ticket is 1 in 41,416,353.

    (a) Since the jackpot is $100 million, our expected value would be:

      100,000,000 / 41,416,353

      = 2.4145052076

    (b) Ah, but wait -- the rules of the lottery state that you get $100 million in annual payments over 26 years. That's not really fair, so let's use the lump sum payment (which everyone takes, anyway). For that, you get about half -- about $50 million. So your expected value is really:

      50,000,000 / 41,416,353

      = 1.2072526038

    Still greater than 1. Ah, but wait...we didn't take into account Uncle Sam. Let's estimate the combined federal and sate taxes at about 45% (California's top bracket is about 9 percent, and federal is about 40, but you can deduct state taxes). So your expected value is really:

      (50,000,000 * (1 - 0.45)) / 41,416,353

      = 0.6639889321

    Now it's much less than 1 -- a sucker's bet. But it gets even worse. We assumed that you were the only one to win. If you have to share the jackpot with another winner, you don't get the full jackpot! What are you chances of having to share the jackpot? Well, it obviously depends on how many other people are playing. Let's say 10 million other people bought 1 ticket each. The chance that at least one of those people won the jackpot is:

      1 - chance all of them lost

      = 1 - (41,416,352 / 41,416,353) ^ 10,000,000

    Blah. Too complicated to compute. Let's instead assume that 1 person bought 10 million tickets (all having different numbers). Then the probability of him winning is:

      10,000,000 / 41,416,353

      = 0.24145052076

    So, your expected value would be:

      (1 - 0.2414505052076) * 0.6639889321 + (0.2414505052076 * 0.6639889321 / 2)

      = 0.7585494948 * 0.6639889321 + 0.2414505052076 * 0.33199446605

      = 0.503668469 + 0.08016023155

      = 0.58382870055

    Now, we didn't take into account the possibility of more than one other person winning, so your expected value is even lower, and of course your expected value will be lower if more people play.

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