A Small Lesson In Probability

By Merreborn

In one of the last issues of the Neopian Times (issue 39), one Neopian suggested a way to win the lottery, guaranteed. I had the same idea once a few years ago (about the real lottery). I asked my father if it would work and he worked through the problem with me. The truth is his method would work. You would win. But it would cost more Neopoints than you would ever win, and would take a very long time. How much would it cost, and how long would it take to buy all your tickets? Let's take a look...

If the NeoPets lottery works like the "real" lottery, you have to match the numbers in the right order. This makes things much harder...

The logical way you would pick your numbers would result in a list of tickets something like this:
Ticket 1 : 1 2 3 4 5 6
Ticket 2 : 1 2 3 4 5 7
Ticket 3 : 1 2 3 4 5 8
...
Ticket 25: 1 2 3 4 5 30
Ticket 26: 1 2 3 4 6 5
Ticket 27: 1 2 3 4 6 7
Ticket 28: 1 2 3 4 6 8
Ticket 29: 1 2 3 4 6 9
...
Ticket 648: 1 2 3 4 30 29
Ticket 649: 1 2 3 4 30 20
Ticket 650: 1 2 3 5 4 6
Ticket 651: 1 2 3 5 4 7
Ticket 652: 1 2 3 5 4 8
...

Notice that after buying 652 tickets, you haven't even changed the first 3 numbers. The list of tickets you need to buy must be very long...

So, let's take a look at this from another angle. In the bottom table, we show a few of the 30 possible selections for the first number. The dashes represent the numbers you haven't picked yet.
1 - - - - -
2 - - - - -
3 - - - - -
4 - - - - -
...
30 - - - - -

For each of those 30 selections, there are 29 possible selections for the next number (29 because you can't pick the same number twice). That's a total of 870 (30 times 29) selections. And you've only picked the first 2 numbers so far!
1 2 - - - -
1 3 - - - -
1 4 - - - -
...
1 30 - - - -
2 1 - - - -
2 3 - - - -
2 4 - - - -

Now, for each of those 870 possibilities, there are 28 possible 3rd numbers. That's 24,360 possibilities for the first 3 numbers!

If we continue this pattern, we see that there are 30 times 29 times 28 times 27 times 26 times 25 possible tickets. That's 427,518,000 (427 million) tickets you'd have to buy! It would cost you over 4 billion Neopoints to buy that many tickets! And even if you had that many NP, you'd have to enter all 427 million tickets. If you could enter a ticket in every second, it would take you over 13 years! Good luck.