Using Bayes Theorem Correctly?

Question

This is not a homework problem, but one from the real world. To keep things g-rated, though, instead of a certain type of criminal I will say Blue Crystal Alien.

It is known that 3% of the population are actually Blue Crystal Aliens (BCA's). Also, 1/3 of all BCA's are left-handed. In the general population, including BCA's and humans, 14% are left handed. If someone is left handed, what is the probability that they are a BCA?

So, I get a prior of .03 times 1/3 equals .01 in the numerator. The denominator is, of course, .14. So, 1/14 ~ 7% chance that any left hander is actually a BCA. This seems really high to me, more than doubling the chance in the gen pop. But I can't see any flaw in my use of Bayes.

Answer 1

I don't see any problems with your calculation.

To convince yourself, try imagining a population of $10,000$. $14$% or approximately one in seven people will be left handed, so about $1,428$ people. $300$ people are a BCA, $100$ of whom are left handed. So the probability that a left handed person will be a BCA is $100/1,428$ which is roughly $7$%.

Answer 2

Indeed.   Your calculations are correct.

$\def\P{\mathop{\sf P}} \begin{array}{rl} \P(B) &= 3/100 \\[2ex] \P(L) &= 14/100 \\[2ex] \P(L\mid B) &= 1/3 \end{array}\qquad\qquad\begin{array}{|rl} \P(B\mid L) & = \dfrac{\P(L\mid B)\P(B)}{\P(L)}\\[1ex] &= \dfrac{1/3\cdot 3/100}{14/100} \\[1ex] &= {1}/{14} \end{array}$

If 1% of the population are left-handed blue crystal aliens, and 14% of the population are lefthanded, then 1/14 of the lefthanded population are blue crystal aliens.

Blue crystal aliens are rare, but proportionately many more of them are left-handed.