Bayes Theorem probability confusion.

Question

Suppose that we have two variables A & B (each one takes the value 1 or 0).

We also know that:

P(A | B=0) = 0.37
P(A | B=1) = 0.17
P(B=0) = 0.29

We need to calculate P(B=1 | A=1).

I guess that initially we have to find that P(B=1) = 0.71 However i need somehow to deduct that P(A) to use Bayes formula! Any inputs much appreciated!

Answer 1

$$\Pr[B = 1 \mid A = 1] = \frac{\Pr[A = 1 \mid B = 1]\Pr[B = 1]}{\Pr[A = 1]}.$$ We also have, by the law of total probability, $$\Pr[A = 1] = \Pr[A = 1 \mid B = 0]\Pr[B = 0] + \Pr[A = 1 \mid B = 1]\Pr[B = 1],$$ and of course since $A$ and $B$ are both Bernoulli, we also have $$\Pr[B = 1] = 1 - \Pr[B = 0] = 1 - 0.29 = 0.71.$$ Thus $$\Pr[A = 1] = (0.37)(0.29) + (0.17)(0.71),$$ if by $$\Pr[A \mid B = 0]$$ you actually mean $$\Pr[A = 1 \mid B = 0],$$ for example. The rest is straightforward.