Friday, October 28, 2016

A Probability Riddle

Some flu strains can jump from people to birds, and, perhaps vice-versa.

Suppose \(A\) is the event that there is a flu outbreak in a certain community say in the next month, and let \(P(A)\) denote the probability of this event occurring.    Suppose \(B\) is the even that there is flu outbreak among chickens in the same community in the same time frame, with \(P(B)\) being the probability of this event as well.

Now let's focus in on the relative flu risk to humans from chickens.  Let's define this risk as
\[R_h=\frac{P(A|B)}{P(A)},
\]
If the flu strain jumps from chickens to people, then the conditional probability, \(P(A|B)\) may well be higher than baserate, \(P(A)\), and the risk to people will be greater than 1.0.

Now, if you are one of those animal-lover types, you might worry about the relative flu risk to chickens from people.  It is:
\[
R_c=\frac{P(B|A)}{P(B)}
\]

At this point, you might have the intuition that there is no good reason to think \(R_h\) would be the same value as \(R_c\).  You might think that the relative risk is a function of say the virology and biology of chickens, people, and viruses.

And you would be wrong.  While it may be that chickens and people have different base rates and different conditions, it must be that \(R_h=R_c\).  It is a matter of math rather than biology or virology.

To see the math, let's start with the Law of Conditional Probability:
\[
P(A|B) = \frac{P(B|A)P(A)}{P(B)}.
\]

We can move \(P(A)\) from one side to the other, arriving at
\[
\frac{P(A|B)}{P(A)} = \frac{P(B|A)}{P(B)} .
\]

Now, note that the left-hand side is the risk to people and the right hand side is the risk to chickens.

I find the fact that these risk ratios are preserved to be a bit counterintuitive.  It is part of what makes conditional probability hard.