Let's suppose event $A$ has $5\%$ chances of happening. The chances that event $A$ is of type $B$ is $70\%$. Event $A^c$ has $5\%$ chances of being of type $B$. Given that event is of type $B$, what is the probability that it is event $A$.
I solved this by using conditional probability formula: $$P(A|B) = \frac{P(A \cap B)}{P(A \cap B)+P(A^c \cap B)} = \frac{\frac{70}{100}}{\frac{70}{100}+\frac{5}{100}}= \frac{14}{15}$$
Am I missing something or why are we given $P(A)=5\%$ for this problem?
That is not correct. $ \small P(A \cap B)$ also needs to consider the probability of event $A$. $ \small 70/100$ is in fact $ \small P(B \mid A)$. Same applies to the denominator.
$ \displaystyle \small P(A \cap B) = P(B \mid A) \cdot P(A) = \frac{70}{100} \cdot \frac{5}{100}$
$ \displaystyle \small P(B) = P(B \mid A) \cdot P(A) + P(B \mid A^c) \cdot P(A^c)$ $ \displaystyle \small = \frac{70}{100} \cdot \frac{5}{100} + \frac{5}{100} \cdot \frac{95}{100}$
Now we apply the formula, $ \displaystyle \small P(A \mid B) = \frac{P(A \cap B)}{P(B)}$