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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