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# Playing Mega Millions for Fun or Profit

Date: 2009/08/27 14:20:00
\$Date: 2009/10/31 21:03:23 \$ \$Revision: 1.7 \$ - Corrected 2009 SS Taxes

## 1.0 Introduction

```The Mega Millions jackpot is currently \$325MM, an enormous sum of
money.  The lottery drawing for this pot is Friday, on 28-Aug-2009.
Is it time to play?

What is your \$1 worth when you pay for that ticket?

According to http://www.megamillions.com/, the first prize is
allocated 63% of the total jackpot, or \$204.75MM.

How much of that \$204.75MM would you take home if you won?

Assuming you are single, living in New York City, and have no other
source of income in 2009, you will need to pay the following taxes
on \$204.75MM of income:

Federal tax: 35%

(8350-0) * 0.10 +
(33950-8350) * 0.15 +
(82250-33950) * 0.25 +
(171550-82250) * 0.28 +
(372950-171550) * 0.33 +
(204750000-372950) * 0.35 = \$71,640,183.50

NY State tax: 6.85%

(8000-0) * 0.0400 +
(11000-8000) * 0.0450 +
(13000-11000) * 0.0525 +
(20000-13000) * 0.0590 +
(204750000-20000) * 0.0685 = \$14,024,978.00

NY City tax: 3.68%

(1706-0) * 1.0000 +
(204750000-50000) * 0.0368 = \$7,532,960.00

Medicare: 1.45%

(204750000) * 0.0145 = \$2,968,875.00

Social Security: 6.2%

(106800) * 0.062 = \$6,621.60

Total Taxes: 50.1%

Federal          \$71.640MM +
NY State         \$14.025MM +
NY City           \$7.533MM +
Medicare          \$2.969MM +
Social Security   \$0.007MM = \$102.584MM

After 50.1% of taxes, the winner of the \$204.75MM first prize
(from the total \$325MM jackpot), will take home \$102.17MM
(\$204.75MM - \$102.58MM).

\$102.17MM is still a nice pay day.  What're the odds of winning?

First prize 1:175711536

To guarantee the prize, you need to purchase 175711537 tickets,
single ticket) to 175711536:175711536, or more simply 1:1.

At \$1 per ticket, you will pay \$175.71MM to win \$102.17MM.  You will
pay \$73.5MM more than you win.

Each ticket purchased "loses" nearly 3/5ths of its value.  Every \$1
spent on this Friday's Mega Millions is worth 58 cents.

Can you increase the value of your dollar?

If you purchase every combination of number drawings, then you must
pay for each of those tickets.  But, knowing that the effective
winnings are \$102.17MM, you may want to bet less than that amount.
Betting \$100MM only gives you a 2.17% return and only if you win it.
Betting \$1MM gives you a 102% return, but the odds are only 0.6%.
Are there classes of numbers that you can avoid to help improve the
value of each \$1 bet?

I suspect that most people play quick draw, which randomly chooses
your numbers.  Presumably, this is fair, since robots do not cheat.

Everybody else choose their numbers by hand, often picking lucky
numbers like 7 or 8, or lucky months (1-12) and days (1-31) of their
favorite birth dates or anniversaries.  This implies that the list
of chosen numbers is heavily weighted toward the smaller numbers.

Choosing larger numbers between 32-56 will not increase your chance
of winning, but should reduce your chance of sharing the pot with
the majority of players who consistently choose the same date or
lucky numbers each week.

There are cognitive biases as well that influence the numbers that
people choose.  Despite the inherent fairness of the draw, our gut
may tell us that the set [ 51 52 53 54 55 56 ] is far more unlikely
than a "more randomly" distributed set such as [ 4 8 15 16 23 42 ].
It may be strategic to choose such patterns, since many people may
try to avoid them.

Since the lottery commission does not release such statistics, these
ideas cannot be quantifiably justified.  The value of such ideas
decrease as more people utilize them, knowingly or not.

Regardless, your chance of sharing the pot winnings with others will
increase significantly with the publicity of a large pot, and the
large number of tickets drawing for the same first prize.

How large should the jackpot be to play?

If you are somehow guaranteed not to share the pot if you win it, the
jackpot needs to exceed \$607MM (\$175.7MM/.499/.58) for your \$1 bet to
not lose value, or to be worth at least \$1.  You are making a nearly
fair bet when the pot is \$607MM, similar to betting \$1 on a fair coin
flip in order to win \$2.

As the jackpot grows larger than \$607MM, the expected value of your
\$1 ticket increases.  But, this still assumes that you've somehow
guaranteed that nobody will share the pot with you.

So, should we play this Friday's Mega Millions for a profit?

Probably not.  Once your \$1 exchanges hands for this Friday's Mega
Millions ticket, it "loses" \$0.42 of its value.  This is like paying
\$1 for a coupon redeemable for only \$0.58.  We are left with playing
just for fun; maybe you will get lucky.