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Of Crabs and Beetroot: A Statistician’s Tale

How probabilities can affect even the smallest business decisions.

So you’ve just sat down to a seafood lunch in order to impress a time-savvy client.

You hate seafood, but there are three dishes that you could really dig into: A large aioli and chicken soup (meal A); a beetroot and sausage risotto (meal B), and the crab (meal C). If only you’d been able to choose that Thai place around the corner.

The waiter approaches: your client is, quite naturally, having fish. You’re just about to order when a (quite junior-looking) waiter, tells you “I’m terribly sorry, but only one of those is good today, but we.. um.. are not sure which one”. Only one? Ridiculous! The client flashes you some “hurry up and just make a choice” eyes. Okay, it’ll be the soup, you say, hedging your bets. Just then, an even more junior-looking waiter interjects, “Chef says the crab is one of the bad ones”.

But then doubt comes over your mind: what if you chose the wrong one of the remaining two? Should you switch choices?

Surely the fact that the crab is off can’t affect the choice you’d already made, can it? I mean, it’s a dish you hadn’t chosen!

Think that’s obvious?

Well, think again: You should switch plates if you want to avoid being unwell in front of your client — the probability that B is good is now 2/3.

Without the extra information, we know that the probability that a given meal is good is 1/3. When the waiter took away option C you still had a probability 1/3 that your original choice was good, but the 1/3 probability that disappeared when C was removed as an option has to go somewhere — and it is now assigned to meal B. What if instead of three there were a million meals, would we switch then?

But that explanation is all a bit vague. Instead, we can look at this through the rigorous eyes of probability theory. At school you were probably taught probability and statistics as a disjointed collection of methods and rules; it turns out a much more unified option exists. This option is Bayes’ theorem, written below in dinner-table format:

P(B is good|C is bad) = P(C is |B is good) x P(C is bad)/P(B is good)

The P(B is good) means “the probability that hypothesis ‘B is good’ is true”, and the P(B is good|C is bad) means “the probability that hypothesis ‘B is good’ is true given if we know data ‘C is bad'”.

Using this equation and throwing in the raw numbers, one can again confirm that you should switch choices, but Bayes’ theorem is much more important than that. In principle it can be used for just about any inferential problem, no matter the sample size and no matter the decision to be made. This is important when the options are no longer as simple as our story.

Bayes’ theorem is a vital cog in inference, and valid inference is a fundamental route to truth in any walk of life: business is no different.

Without valid inferences, how can you tell that the changes you made to your business are really effective? How much of your growth is due to the approaches you’ve just implemented, and how much just due to changes in the market? At Black Swan, inference forms a crucial part of our work — it’s vital that clients avoid the traps in decision-making that coincidences can present, and the losses in revenue that bad inferences bring.