What is Bayes’s Theorem?
In: 48
According to google, 87% of billionaires are male.
Gosh! That means 87% of males must be billionaires, right?
Wrong.
Bayes’ theorem tells you precisely *how* wrong.
The probability of being male *given that* you’re a billionaire is 87%. You can write that as P(male|billionaire) = 0.87
What you want to know is the reverse of that: P(billionaire|male), the probablility you’re a billionaire *given that* you’re male.
And believe me, that number is not 0.87, or anywhere close to it.
But if we have two more bits of information, we can work out what that number is.
We need to know P(male) and P(billionaire), independent of each other. What percentage of all people are male, and what percentage of all people are billionaires.
Off to google:
* Total number of humans = 8,036,677,056
* Total number of billionaires = 3,311
* Percentage of humans that are male = 50.42%
* Percentage of billionaires that are male = 87%
Bayes’ theorem says that
P(A|B) = P(A) * P(B|A)
—————
P(B)
Translating that into our terms, that’s
P(billionaire | male) = P(male | billionaire) * P(billionaire)
—————————————-
P(male)
P(male | billionaire) = 87% / 100 = 0.87
P(billionaire) = 3,311 / 8,036,677,056 ≈ 0.000000412
P(male) = 50.42% / 100 ≈ 0.5042
Now we can plug all those numbers in:
P(billionaire | male) = (0.87 * 0.000000412) / 0.5042 ≈ 0.000000711
The chance of being a billionaire if you’re male is thus 0.0000711%.
That’s one helluva long way from 90% – and an awful lot of racism and bigotry is based on not understanding the difference.
If you want to know *why* Bayes’ theorem works, someone’s explained it a helluva lot better than I can, using Venn diagrams.
[Take a look](https://oscarbonilla.com/2009/05/visualizing-bayes-theorem/)
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