Probability of success and failure

On average for $14900$ transactions:

$100$ are suspicious, and of these $92$ are identified as suspicious;

and

$14800$ are legitimate, and of these $1184$ are (mis)identified as suspicious.

So out of $1276$ transactions identified as suspicious, $1184$ are in fact legitimate (on average).

By the way, this agrees with Graham Kemp, but is perhaps a bit more intuitive.

The incidence of a suspicious transaction in a bank is 1 in 149. They are able to correctly identify a legitimate transaction 92% of the time. However, this bank is also able to correctly pinpoint a suspicious transaction 92% of the time. One day, the bank identify a transaction as suspicious. What is the exact probability of the transaction actually being legitimate?

Let $S$ be the event that a transaction is suspicious.   Let $T$ be the event that a transaction is identified as suspicious.

We are given: $\mathsf P(S)=1/149, \mathsf P(T^\complement\mid S^\complement)=0.92=\mathsf P(T\mid S)$.

We seek, by means of Bayes' Rule: $\mathsf P(S^\complement\mid T)~{=\dfrac{\mathsf P(T\mid S^\complement)\mathsf P(S^\complement)}{\mathsf P(T\mid S)\mathsf P(S)+\mathsf P(T\mid S^\complement)\mathsf P(S^\complement)}\\ = \dfrac{0.08\cdot148/149}{0.92\cdot1/149+0.08\cdot148/149} \\ =\dfrac{296}{319} }$

Let $S$ denote the event "the transaction is suspicious", and $I$ denote the event "the transaction is identified as suspicious"

What the question is asking for is $$P(\neg S| I)$$ (since $\neg S$ is the event "the transaction is not suspicious", i.e. "the transaction is legitimate").

Now, we use good old Bayes to get

$$P(\neg S|I) = \frac{P(I|\neg S)\cdot P(\neg S)}{P(I)}$$

You already know that $P(I|\neg S) = 0.08$, and that $P(\neg S) =1-P(S) = 1- \frac{1}{149}=\frac{148}{149}$.

So, what is the probability of $I$? Well, for that, we use the law of total probability:

$$P(I) = P(I|S)\cdot P(S) + P(I|\neg S)\cdot P(\neg S) = 0.92 \cdot \frac{1}{149} + 0.08\cdot \frac{148}{149}$$

Can you take it from here?