\[ P(A | B) = \frac{P(B | A) * P(A)}{P(B)} \]
In other words:
It states that if we have partitions (\(B_{1}\) … \(B_{k}\)) of a sample space (which have non-zero probabilities), then for any event A (event also has non-zero probability) in the sample space:
\[ P(B_{i} | A) = \frac{P(B_{i} \cap A)}{\sum_{j = 1}^{k} P(B_{j} \cap A)} = \frac{P(A | B_{i}) * P(B_{i})}{\sum_{j = 1}^{k} P(A | B_{j}) P(B_{j})} \]