## Numbers Sometimes Lie

(Submitted by Skepticality listener  Stephen Hayko.)

I do clerical work for a company that uses part numbers that are six digits long and begin with either a 5 or 6. When we order parts, our ordering system generates a purchase order (PO) that is six digits long and sequential.

We’ve been using this ordering system for about a year, and throughout the company, we typically place about 45-50 orders in the system every week, in my branch. We’re one of 25 branches in the US that uses this system, and we are one of the higher-volume branches – most other branches use about 30-35 orders per week.

In March, I placed an order for part number 649384. This is a relatively common piece and we typically sell 8-10 of this part per week – so it accounts for 16-20% of our orders. Lo and Behold! The PO was 649384.

Given that information, what are the odds that PO 649384 was attached to an order for part number 649384?

Thanks!

Below are the extended notes provided by contributing editor Mark Gouch for use in Skepticality Episode 253. Mark is a wastewater treatment system operator and engineer living in Smithtown, NY (Long Island). He started to become interested in coincidences after recognizing the series of events that conspired to get him employment on Long Island many years ago. Two of Mark’s recommended books include “The Drunkard’s Walk: How Randomness Rules Our Lives” by American physicist and author Leonard Mlodinow, and “The Hidden Brain: How Our Unconscious Minds Elect Presidents, Control Markets, Wage Wars, and Save Our Lives” by Shankar Vedantam.

Take a look and leave your comments below. Also, please be sure to listen to the podcast for our own hilarious commentary.

This problem seemed very straightforward at first, but on closer review it seems that there is something interesting hidden in the details Steve provided. Estimating the total number of POs generated company-wide using the average of the ranges you gave comes to about 780 POs per week. That’s about 3,586 per month, and 43,030 annually. Steve said the numbers generated automatically are six digits long, and are sequential. So if 43,030 are generated annually, it would take 649,384 / 43,030 years to hit number 649384, or about 15 years, one month. So barring any large increase or decrease in business, in about 15 years you may see this happen again. But wait, Steve also stated that the company has been using this system for a year. Something is fishy here. If the numbers are sequential, and they’ve used this for a year, then they must not have started at 000 001. They must have started somewhere around 649,384 – 43,030 = 606,354. That is, if the numbers Steve gave were close to correct. Starting to wonder if this is some sort of trick question here Steve. Something does not add up. Literally.

So either Steve submitted a trick question which he knows is impossible, or someone, for some reason, decided to covertly tamper with the automatic PO number generating software to make it start at some number other than 000001. Perhaps someone thought PO numbers like 000001, 0000002, etc. would make the company look like a startup, or just would look odd. PO number 606354 makes the company look like they’ve been in business for a long time, and/or process quite a lot of POs. So this great mystery deserves some investigation. Inquisitive minds want to know what was the first PO number generated, who determined what that number was, how did they determine it, and why? And was it part of a conspiracy, or did this mysterious person act alone? A reasonably thorough investigation is certainly in order. There must be a logical explanation.

A number starting in 60 does not look like someone used their birthdate, which would be weird anyway. Does Steve know the last six digits of the CEO’s social security number? Well, there could be a mundane explanation, like the numbers were sequential for many years, maybe kept on a clipboard or something, and only a year ago was it computerized. Let’s go with that, and forget the conspiracy theory. In fact, everyone please forget all conspiracy “theories.”

So back to the actual question. It seems that Steve already knows the answer to his question. He said that this common part accounts for 16 to 20% of their orders. So the odds of any one order having this part number on it should be approximately……16 to 20%! Grab any random order out of the pile (or computer system) and there will be a 16-20% chance that it has this part number on it. That goes for any PO number: 650000, 700000, 131313, and also for 649384. Steve knew the answer; he just did not know that he knew. This is certainly not a criticism. It is better to not know that you know something than to think you know something that you do not. The fallacy was that he thought the odds would be different for that one special PO number, but they are not. The odds are the odds. The odds, in this case, are perfectly rational – but not sequential.

## George Hrab on Coincidence

Last week we were excited to learn that George Hrab mentioned us in episode 251 of the Geologic Podcast. We’re definitely fans of his wide range of work, so the shoutout was a personal moment for the team. Some of us were even mildly verklempt, which was all the more relevant thanks to his mention of Gefilte fish, though less so since we’re not actually Jewish.

After a brief conversation with George via email, he graciously provided us with permission to post a transcript of his thoughts on the subject which I’ve placed below, followed by some additional thoughts by me, assuming you care. Please validate me by caring. Also, please listen to the podcast if you haven’t already since you get the nuances of George’s delivery, along with his general Georgeness.

Geologic Podcast #251 – Coincidence Transcript

I saw an interesting web site–no, a little blog post. There’s a place called The Odds Must Be Crazy. We’ll try to link to that in the show notes. But someone went onto The Odds Must Be Crazy–Brian H–he wrote this. He said, “I was listening to George Hrab’s podcast (episode 240) on my iPod while heading out to one of my familiar lunch spots in Santa Monica, California. In this episode George did a bit called the History Chunk where he tells what happened on this particular date in history, usually in chronological order, and the makes some kind of joke about it. He mentions how in 1982, boxer Duk Koo Kim died after a bout with Ray ‘Boom Boom’ Mancini. Thirty seconds later I see Ray ‘Boom Boom’ Mancini having lunch in the very restaurant I was walking into.  I clandestinely snapped his picture.”

This site is really interesting, and it talks about sort of the odds of things happening and how it can seem that the odds of something must be so astronomical that there must be some kind of a sign. So this Brian was listening to the show, I say “Boom Boom” Mancini, he looks up, and there’s “Boom Boom” Mancini. Now how could we calculate the odds of that occurring? I don’t know, but they’re astronomical. They’re astronomical. And yet if you think, “how many people that listen to the show didn’t see Ray ‘Boom Boom’ Mancini when I said it?”, that would help to demonstrate the odds being not quite as horrifically set against as you might imagine.