# Probability of success and failure

I've an interesting question on my hand which I've approach several others but all of them gives me different insights to this probability question.

Here it is,

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?

From my personal point of view, if the question ask for the probability of the transaction actually being legitimate, states that the rate is 148⁄149. The bank is able to correctly identify (which they fail to) a legitimate and suspicious transaction. Therefore, the failure % should be (8% * 8%) which is 0.08 * 0.08 = 0.0064. Hence, the probability of it actually being legitimate is 148⁄149 * 0.0064 = 0.00636.

However, i asked various people of their opinion and some states that the probability should be just 148⁄149* 0.08.

Therefore, what should be the most probable answer to problems like this.

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: $\mathsf P(S^\complement\mid T)~{=\dfrac{\mathsf P(T\mid S^\complement)\mathsf P(S^\complement)}{\mathsf P(T\mid S)\mathsf P(S)+\mathsf P(T\mid S^\complement)\mathsf P(S^\complement)}\\ = \dfrac{0.08\cdot148/149}{0.92\cdot1/149+0.08\cdot148/149} \\ =\dfrac{296}{319} }$

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.

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?