A Real Lottery

Five people go to lunch (Well, actually, five computer geeks go to lunch.)

The Mega Millions Lottery topic comes up in conversation. They decide to purchase a joint lottery ticket.  The rules for choosing numbers for this lottery are:

1. Pick five numbers, without replacement,
   from the set {1,2,3, . . . 56}.
2. Then pick one number
   from the set {1,2,3, . . . 46}.

A suggestion is made that all the choices be prime. Since there are sixteen prime numbers between 1 and 56 and thirteen primes between 1 and 46 then the following process is decided on:

Each person picks their favorite number from 1 to 16 and then the group picks a common favorite number from 1 to 13. Once the numbers are selected they are paired with the prime numbers and the set of numbers for the lottery ticket is generated.

The worksheet for picking lottery ticket numbers has room for five sets of numbers, and the price of a lottery ticket is $1.00. For this reason the group decides to collect $1.00 from each person instead of $0.20 and thus the group can fill out the entire worksheet.

This decision forces the group to either reuse the same five prime numbers or to come up with four more sets of numbers. Since the odds of winning will be increased if four more sets of numbers are generated then that is the route decided on. It is decided that some sort of transform of the original set of numbers should be applied to get the rest of the sets. Here is the algorithm chosen:

1. Increment each number by one getting a new set of six even numbers.
2. Select the first three primes and the last three even numbers to get the next set.
3. Select the first three even numbers and the last three prime numbers to the the next set.
4. Finally apply the Collatz Conjecture Algorithm to each of the original numbers noting the length of the resulting sequence (modulo 56 or 46) and use those lengths for the final set.

NOTE: Here is a link to a more humorous take on the Collatz Conjecture

The five favorite numbers were: {9, 13, 11, 3, 16}, and the common favorite number was 7.

When you match up the first sixteen prime numbers with the group's favorite numbers you get the following:

( 01, 02 )
( 02 ,03 )

( 03, 05 ) A Favorite - Collatz Path Length = 05

( 04, 07 )

( 05, 11 )

( 06, 13 )

( 07, 17 ) A Favorite - Collatz Path Length = 16

( 08, 19 )

( 09, 23 ) A Favorite - Collatz Path Length = 15

( 10, 29 )

( 11, 31 ) A Favorite - Collatz Path Length = 50 = 106 MOD 56

( 12, 37 )

( 13, 41 ) A Favorite - Collatz Path Length = 53 = 109 MOD 56

( 14, 43 )

( 15, 47 )

( 16, 53 ) A Favorite - Collatz Path Length = 11

Here are the resulting five sets of numbers.

• {05, 17, 23, 31, 41, 53} Original 
• {06, 18, 24, 32, 42, 54} Even
• {06, 18, 24, 31, 41, 53} Even Then Odd
• {05, 17, 23, 32, 42, 54} Odd Then Even
• {05, 12, 15, 50, 53, 11} Collatz Path Length

Here are some numbers:

It was estimated that $1,500,000,000.00 spent on the mega millions lottery.

If you choose without replacement and without regard of order five numbers from the set {1, 2, 3, . . . 56} the number of different choices comes out to be the binomial coefficient "56 choose 5" which is 3,819,816.

If you choose one number from the set {1, 2, 3, . . . 46} the number of choices comes out to be 46.

Taking both together the number of possible choices is
46 * binomial(56,5) = 175,711,536


What this means is that the five sets of numbers above have a five in one hundred seventy five million seven hundred eleven thousand five hundred thirty six chance of being chosen. Another way to say it is that the odds are: 1 in 35,122,306


So here are the expected results:

And Here are the Actual Results

As you can tell the expected results in all five instances do not match the actual results. And, you can also tell we messed up the Collatz path lengths for some of the numbers.  And, if we had done them correctly, we still would not have successfully predicted the numbers that were ultimately selected.

Here are some more numbers that fall into the category of "What If".  And really that is what the lottery is all about (what if).

An interesting 'bottom-line' question now becomes:

Just how fast do you think you could you 'burn-through' sixteen million one hundred and twelve thousand two hundred and fifty dollars? 

30 Mar 2012 Mega Millions Total Prize
Total	Per Yr / 26 Yrs	Lump Sum
$656,000,000.00	$25,230,769.23	$462,000,000.00
If one of our numbers had won then the total would have been split four ways.
Total	Per Yr / 26 Yrs	Lump Sum
$164,000,000.00	$6,307,692.31	$115,500,000.00
Next each of us would have received one fifth of the one forth of the total.
Total	Per Yr / 26 Yrs	Lump Sum
$32,800,000.00	$1,261,538.46	$23,100,000.00
Next the federal government gets a 25% cut (taxes) leaving each of us with 75% of of our 20% of our 25% of the total
Total	Per Yr / 26 Yrs	Lump Sum
$24,600,000.00	$984,000.00	$17,325,000.00
Finally the state of North Carolina gets a 7% cut (taxes) leaving each of us with 93% of our 20% of our 25% of the total.
Total	Per Yr / 26 Yrs	Lump Sum
$22,878,000.00	$879,923.08	$16,112,250.00