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Are you better off playing the lottery or saving
the money? Assume you can buy one ticket for $5, draws are made monthly, and a
winning ticket correctly matches 6 different numbers of a total of 49 possible
numbers.
The probabilities: In order to win, you must pick all
the numbers correctly. Your number has a 1 in 49 chance of being correct. Your
second number, a 1 in 48 chance, and so on. There are exactly 49 x 48 x 47 x 46
x 45 x 44 = 10,068,347,520 ways to pick 6 numbers from 49 options.
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Are you better off playing the lottery or saving
the money? Assume you can buy one ticket for $5, draws are made monthly, and
a winning ticket correctly matches 6 different numbers of a total of 49
possible numbers.
The probabilities: In order to win, you must pick
all the numbers correctly. Your number has a 1 in 49 chance of being correct.
Your second number, a 1 in 48 chance, and so on. There are exactly 49 x 48 x
47 x 46 x 45 x 44 = 10,068,347,520 ways to pick 6 numbers from 49 options.
But the order in which you pick them does not
matter, so you actually have a few more ways to win. You can pick 6 different
numbers in exactly 6 x 5 x 4 x 3 x 2 x 1 = 720 orders of choice. Any one of
those orders would still win the lottery.
Putting this all together, your ticket has
720/10,068,347,520 = 1/13,983,816 chance of winning. This equates to a
.000000071 percentage chance.
If you played one ticket every month from age 18
to age 65, you would have 47 x 12 = 564 plays. Your odds of not ever winning
would be calculated using a binomial distribution to be .9999599568, meaning
your chances of winning would be 1 â .9999599568 = .0000400432.
So, if the lottery winnings averaged $10 million
over this time period, your expected return would be less than .0000400432 x
$10 million = $400.43.
(It’s less than $400.43 because your 564 plays
are spread out over the next 47 years, so the present value of these future
plays would be significantly less than if you were able to play all 564
immediately. The $400.43 assumes you play all 564 plays today, which makes it
the highest possible expected value.)
REQUIRED:
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A.
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What would your $400.43 be worth if you invested
it at 1% real interest for 47 years?
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B.
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If, instead, you wrote down your 6 numbers on a
piece of paper, and deposited your $5 in a bank at 1% real interest, how much
would you have at the end of the first year?
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C.
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If you did this every year for 47 years, how much
would you have at age 65?
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D.
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If you earned 5% real interest on your deposits,
how much would you have at age 65?
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