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One of the most important notions in probability and statistics is Bayes’ Theorem, and it can be a little difficult to understand.It involves negotiating probabilities of conditional events to find out what is really going on.
Here’s how it goes:
Suppose you are living with a partner and come home from a business trip to discover a strange pair of underwear in your dresser drawer. You will probably ask yourself: what is the probability that your partner is cheating on you?
In order to figure this out probabilistically, you’ll need to estimate a few probabilities.
The first, “y” is the probability that the underwear appeared because you are in fact being cheated on.
If he’s cheating on you, it’s certainly easy enough to imagine how the panties got there. Then again, even (and perhaps especially) if he is cheating on you, might expect him to be more careful. Let’s say the probability of the panties appearing, conditional on him cheating on you, is 50 per cent
Next, you want to estimate “z,” the probability that the underwear appears but he’s not cheating on you – maybe they’re a gift, or a platonic female friend stayed over and left them, or there was a luggage mix up. Silver says that you could put the probability at 5%
Finally, you need to figure out the prior probability, or “x”. This is the probability that, had you not found the underwear, you think your partner is cheating on you.
Studies have found, for instance, that around 4 per cent of married partners cheat on their spouses in any given year, so we’ll set that as our prior.
Bayes theorem computes the posterior probability, or the probability that, given you found the underwear, your spouse is cheating.
The posterior probability is equal to:
xy/[xy + z(1-x)]
In this case 29%, which is still fairly low.
The book goes back to Bayes’ theorem constantly, and for excellent reasons – it’s an exceptionally powerful way to honestly gauge a complex reality based on estimable probabilities, and is perhaps the most important theory in modern probability.
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