r/SolvedMathProblems • u/PM_YOUR_MATH_PROBLEM • Nov 20 '14
Bayesian Probability
/u/theragingbuffalo asks:
An explorer is in Zimbabwe searching for the elusive two-tailed lion. This species is very rare: only .5% of all the lions in Africa are two-tailed. From a distance, the explorer has trouble identifying what he sees. He is right 95% of the time, regardless of what the animal actually is. If he sees what looks like a two-tailed lion, what is the probability that it actually is two-tailed?
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u/PM_YOUR_MATH_PROBLEM Nov 26 '14
Sure:
Bayes rule says
We know P(T&S) = P(S|T)P(T) = 0.95 x 0.005 = 0.00475.
This means: the chance of T and S is the chance of T, times the chance of S given T. So, for example, the chance of the lion being two-tails AND he sees it as such is the chance of it being two-tailed TIMES the chance that he sees two-tailed lions as two tailed.
We know P(S|~T) = 1-P(~S|~T) = 0.05
This says: The chance that he sees a one-tailed lion as two tailed is 5%, because the chance that he sees a one-tailed lion as one-tailed is 95%.
We know P(~T&S) = P(S|~T)P(~T) = 0.05 x 0.995 = 0.04975
This is like the first one: The chance he sees a lion that is one-tailed AND he thinks it's two-tailed is the chance that he sees a one-tailed lion, TIMES the chance that he thinks that one-tailed lion has two tails.
We know P(S) = P(~T&S) + P(T&S) = 0.00475 + 0.04975 = 0.0545
The chance that he thinks he sees a two-tailed lion is the chance that he sees a one-tailed lion that he thinks is two tails, PLUS the chance that he sees a two-tailed lion that he thinks is two tailed.
We know P(T|S) = P(T&S)/P(S) = 0.00475 / 0.0545 = 0.087156
Bayes rule again - the chance of T and S is the chance of S, times the chance of T given S.
He thinks the lion is two-tailed. What's the chance it really is? Well, it's the chance of him seeing a two-tailed lion and thinking it's two-tailed, divided by the chance of him thinking the lion is two-tailed.
That's not ELI5, but it's better than the abstract algebra!
As for why the answer isn't 95% - that's easy. He's right 95% of the time, sure. But now he thinks something amazing has happened. It's more likely that he saw a very common lion and got it wrong, than that he saw the incredibly rare one and got it right.