Beauty & Intelligence

Here is a fun probability result which has to do with conditional probability. Its not very hard to see what's going on but its a fun result nonetheless. I just made up a dummy example to make it more fun :)

Lets take 2 fairly independent attributes like beauty and intelligence (It can be argued that they are not independent by appealing to genetics and preferential/unequal selection rights for species perpetuation but lets ignore that for now and think simple). Lets assume beauty and intelligence are independent.
i.e if
B=beautiful
I=intelligence

P(I)=P(I|B) ->(1)

Now, think of all the people who stick in your memory (as opposed to you forgetting them after a few days of meeting them). Lets assume (simplistically)
that the people who stick in your memory are ones who are either intelligent or beautiful.

Lets throw in some numbers.
Lets assume the prior probability of someone being beautiful is
0.4
and for someone being intelligent is 0.1
Lets assume 10% of the beautiful people are intelligent.

Say, you've met 200 individuals in your lifetime.

If we go with the assumption of you being able to recollect only people who
are intelligent or beautiful, you will remember
80+12=92 individuals

Now when you look at this sample set, lo and behold, it looks like intelligence
and beauty are negatively correlated!

This looks counterintuitive since it looks like,
P(I)=0.21
but P(I|B)=0.1
(thus violating the independence assumption in (1))

whereas in reality, the conditional probability eqns are,
P(I|B,P) < P(I|P)
where P=I U B
which is another case of selection bias at work..

Hmmm..I wonder how many people feel this way about beauty and intelligence ;)