Wednesday, August 14, 2013

Just how much money does Walter White have? A math problem

Spoiler alert: Breaking Bad spoilers below.

At the end of the last season of Breaking Bad, Skyler presented Walter with an interesting math problem. In a storage shed, probably somewhere in Albuquerque's War Zone (roughly from San Mateo to Louisiana, south of Menaul), she had an enormous pile of Walt's money, and she couldn't begin to figure out how to count it.

Like many psychopaths, Walter is a smart guy; I certainly couldn't cook up a batch of 99% pure crystal meth (I mean, I've never tried, but I never even took chemistry in college). Skyler's no dummy either, her refusal to ditch Walt notwithstanding. She is certainly good at math, since she is a licensed accountant. So I'm amazed that neither of them could even make a good estimate of the amount of money they had.

Skyler was sensible enough to think of counting by weight. Her problem was that the bills were in different denominations, so even if she knew how many bills she had, she wouldn't know how much that was. But this is a relatively trivial problem, if one applies a little basic statistics, something both Walter and Skyler should be familiar with.

I can't weigh Walter's money of course. I'd be happy to try, and if the pile ended up a few stacks of hundreds short, I can hardly see how he'd notice. But I can estimate by volume.

Walt's pile appeared to be a rectangular prism of about 6' long by 4' wide by 4' high, for a volume of 96 ft^3. The volume of a stack of 100 bills of any denomination is about 6.45 in^3(1) or about 0.00373 ft^3 (That seems small, but remember we are talking about volume so we are dividing by 12^3 cubic inches=1728in^3).

Simple division tells me that Walt has about 25,000 stacks of bills in there, or about 2,500,000 bills. If they were all hundreds, he'd have a quarter billion dollars! I assume Walt's not keeping any denomination less than a twenty, so bare minimum he'd have 50 million bucks.

But they could do a much better estimate. All they'd have to do is take a random sample of about 100 stacks, and find how many were of each denomination. Say that your random sample of 100 stacks had 40 stacks of hundreds, 50 stacks of twenties, and 10 stacks of fifties. (I assume fifties are least likely, since big drug dealers deal in stacks of hundreds and small-time buyers mostly use twenties.) If your sample is representative, you can say the average denomination is about 100*.4+50*.1+20*.5=$60, meaning that Walt has 150 million bucks in there.

Since Walter estimated that the methylamine he stole from the freight train would be worth about 300 million when cooked, this would mean he's cooked about half of it and sold it, which makes sense. I wonder what he'll do with the other half? He'd probably like Lydia to buy it wholesale, but she is not really enthusiastic about his retirement plan. Maybe those guys that Mike introduced him to want to buy it. But then he'd have to get into what exactly happened to Mike. But, as mathematicians like to say, I'll leave the general case for the next generation.

Bonus question 1: Research the average income of an Albuquerque car wash. Assuming Walt and Skyler bought another car wash as Walt discussed, how long would it take them to launder all that money?

Bonus question 2: Using Jesse's guilt-driven system of distributing his "blood money" by driving through Albuquerque and throwing out a bundle of bills every few houses, how long will it take him to get rid of his five million dollars? How might this impact the Albuquerque economy?


DesertRat said...

Walter's arrangement with Declan & crew was that they would get 35%. You would think he had a similar arrangement with Lydia. So if he has $150 million, then that's 65% of the actual dollar amount. Giving a reasonable margin either side of the $150 million estimate, that would be $200-$250 million, which would mean he's cooked approximately 65-75% of the methylamine.

Anonymous said...

The pile is up to Skylar's knees, there's no way it's 4 feet high. Maybe more like 2 feet.

Jim Stewart said...

@desertrat: Oh! I forgot that he wasn't getting all the money. I should have taken that into account.

@Anonymous: you might be right about the dimensions. I was picturing the pile in my head from memory; I only added the picture at the end.