12
$\begingroup$

Is there a way to resolve probability of an event, given another event that never happens? Mathematically speaking the problem is:

Given that $P(B) = 0$,

$$P(A|B)=\frac{P(A \cap B)}{P(B)} = \frac{0}{0}$$

Is this probability vacuously $0$ of $1$? Can we show that it's one or the other?

  • 1
    you might look at http://en.wikipedia.org/wiki/Regular_conditional_probability2012-02-16
  • 0
    I would think that $P(A|B)$ would be undefined. However, a quick search of wikipedia shows that there are ways to approximate $B$ by events with nonzero probability and consider a limit. (http://en.wikipedia.org/wiki/Conditioning_(probability)#The_limiting_procedure)2012-02-16
  • 9
    Conditioning on zero-probability events is used quite commonly when one is dealing with continuous random variables, since for a continuous random variable $X$, $P\{X = a\} = 0$ for all $a$, while we still want to talk about $P(B|X=a)$ and even use the law of total probability in the form $$P(B) = \int_{-\infty}^\infty P(B|X=a)f_X(a)\mathrm da.$$2012-02-16

3 Answers 3