Powerball! Buy All The Tickets!!

Ok! So, somewhere in the US there is a lottery that now has a jackpot of $1.3 billion. Yes. Billion. As in just enough money to launch a presidential campaign. The web is filled with simulators, “strategies” for winning, basic expected value calculations, slightly more complicated probability questions, etc.

Eventually, at some point, either inspired through a lesson or just a natural evolution of the conversation, we get to the question of guaranteeing a win. There are some 292 million combinations to choose from and to be absolutely certain that one will win the top award, one has to buy all possible combinations (we want to win with 100% certainty). The exact number of combinations, and hence, number of tickets to purchase is $$\frac{69!}{5!64!}\cdot 26 = 292201338$$

Now, I like to have fun with questions like these because there is the academic side of it (just buy all the tickets!) and then there’s the practical side of it — how? It also makes for a great math lesson that is a bit different from the standard, run-of-the-mill probability of ruin types of questions. Those are good questions, but if you’re looking to add a bit more to your lesson repertoire, see below.

Questions

All those tickets!

Ok, so we have to purchase 292,201,338 tickets.

If we had to fill out each ticket by hand, how long will that take?
Additional follow up questions are, “How many days will it take?”, “If we bought now, will we finish in time for the next drawing?” “Where are we going to be able to purchase 292 million tickets?”


If we chose to buy all tickets by the Quick Pick feature, how many tickets do we have to buy?
Remember, Quick Pick chooses tickets randomly! This is the old coupon collector problem. Other logistical problems are, how many liquor stores and other lottery ticket sellers will you have to visit? No single location is carrying hundreds of millions of tickets.


Why can’t we just give the lottery enough money to cover the purchase of all tickets?

So, why can’t we? Actually, I don’t know! Do I have to buy a ticket? It seems like a reasonable solution! I should just be able to go to the lottery commission and say, “Here’s money, let’s now have it known that I have purchased all combinations.” Seems like a win-win. Clifford Pate offers this as a way to secure cash.


We are guaranteed to win if we buy every combination. Are we guaranteed to profit?

If a million people are playing, including you, what are the odds that they all miss and you are the only one to hit?
How many people will you share the jackpot with?


Let’s be realistic here …

Ok, let’s just say that somehow we did in fact, purchase every combination.

Where will we store 292 million tickets?

How many storage boxes is that? How big will the storage facility have to be? How much is it going to cost just to do this? And remember the clock is ticking!! We have to do this in a matter of days! What kind of insurance policy would we have to take out? Who would insure this? (Actually there are companies that insure crazy things like this.)


What kind of quality control process do we need to ensure that we have all the tickets?
How can we ensure that all tickets have been accounted for? What if we lose one? What if it is the winning one? What legal recourse do we have if we can’t find the ticket but we know we bought it? “Trust me, I bought it.” isn’t going to fly.


What kind of security detail will we have to hire?

Can we trust our security detail? I mean after all, the winning ticket IS present! And it’s worth billions! What if we go into our storage facility and find the ticket. This is a $1.3 billion asset. What if the security team colluded and took away our ticket and decided to split the prize among themselves? Money, this amount of money, does strange things. Rational behavior arguments are out the window!

This problem is much like trying to figure out how Santa Claus makes all those deliveries. Enjoying vexing your students!


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