Probability problem - Bayes or Conditional?

Question

I need to solve this exercise and I do not know how to do so.

The only information that I have are:

What is given:

$a,b \in \{0,1\}$

$p(b) = (b+1)/3$

$p(a|b) = (a+b)/(1+2b)$

What is requested:

$p(b| a=0)$ ?

I tried to apply the Bayes formula, however I do not have the $P(a)$. For the conditional probability I do not have the intersection between $a$ and $b$.

Thanks

Answer 1

To get probability of a ${P(a=0)}$, you need to use the law of total probability. ${P(a=0) = P(a=0 | b=0)*P(b=0) + P(a=0 | b=1)*P(b=1)}$.

Once you compute that and have ${P(a=0)}$, you can use Bayes’ rule just like you started to do.

A less sloppy approach would be to first find the formula for ${p(a,b)}$:

${p(a,b) = p(a | b)* p(a) = \frac{(b+1)*(a+b)}{3*(1+2b)}}$

Then,

${p(b | a=0) = \frac{p(0,b)}{p(0,0)+p(0,1)}}$

Now simply plug into the formula for ${p(a, b)}$ that you found using Bayes.