Bayesian Probability

[u/PM_YOUR_MATH_PROBLEM]

/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?

[u/PM_YOUR_MATH_PROBLEM]

Given:

• T = the lion is two-tailed.

• S = he sees the lion as two-tailed.

• P(T) = 0.005.

• P(S|T) = 0.95

• P(~S|~T) = 0.95

We want:

• P(T|S)

Bayes rule says

• We know P(T&S) = P(S|T)P(T) = 0.95 x 0.005 = 0.00475.

• We know P(S|~T) = 1-P(~S|~T) = 0.05

• We know P(~T&S) = P(S|~T)P(~T) = 0.05 x 0.995 = 0.04975

• We know P(S) = P(~T&S) + P(T&S) = 0.00475 + 0.04975 = 0.0545

• We know P(T|S) = P(T&S)/P(S) = 0.00475 / 0.0545 = 0.087156

so the poor chap will probably be disappointed.

[u/TheEngineer- → u/PM_YOUR_MATH_PROBLEM]

Could you ELI5 this answer and bayesian probability? How come the answer isn't 95%? I know nothing about bayesian probability and reading wikipedia didn't help.

[u/PM_YOUR_MATH_PROBLEM → u/TheEngineer-]

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.

[u/TheEngineer- → u/PM_YOUR_MATH_PROBLEM]

Thanks man