Reading what you have said I will try to start again and break it down.

(P1) Probability of transaction being fraudulent = 0.01

(P2) Probability of fraudulent transaction being flagged = 0.99

(P3) Probability of transaction being flagged accurately = 0.01

I assume the 3 possibilities have to be combined in the dependent order so P1*P2*P3 = 0.000009

You are describing P2 loosely and P3 wrongly, which makes it likely that you either are not fully understanding them, or will confuse yourself going forward. Exact wording, and careful rewording, are essential in probability.

As I showed before, you really need to state them explicitly in terms of conditional probabilities, which I am assuming you are familiar with.

Here's the problem data again, quoted exactly with labels added:

A financial institution develops a fraud detection system which is not entirely accurate.

If a fraudulent transaction is detected, the probability that it is flagged (P2) is 0.99.

On the other hand, a transaction is flagged with probability 0.01 when it is not actually fraudulent (P3).

Assume that the probability that a transaction is fraudulent is only 0.01 (P1).

The reality is, as I read it:

(P1) Probability of transaction being fraudulent =

**P(fraudulent)** = 0.01 (correct)

(P2) Probability of fraudulent transaction being flagged =

**P(flagged | fraudulent)** = 0.99 (correct, but unclear wording)

(P3) Probability of

**nonfrauduluent **transaction being flagged

**wrongly **= P(flagged | not fraudulent) = 0.01

What you said for P3 doesn't make sense. If you look through these, you see that they are saying (P1) that there are not many fraudulent transactions; (P2) most of them are caught; (P3) few are wrongly flagged.

You can't multiply these together to get something meaningful; never, ever

*assume *such a thing!

**Answer to question 1** is 0.000009 (overall probability of receiving a fraudulent transaction **that has been** accurately flagged)

No. You are asked:

What is the probability that the transaction is actually fraudulent if it has been flagged?

That means P(fraudulent | flagged), not P(fraudulent

**and** flagged), which is what you say you have calculated. Your calculation is wrong anyway.

**First answer to question 2** is 11,000 * 0.000009 = 0.99 (1 transaction in 11,000 is expected to be fraudulent and accurately flagged)

However as you pointed out, question 2 just asks how many transactions are expected to be flagged irrespective of flag being accurate so I think I don't need to include P3.

Right: You have misinterpreted the question. And the probability you need to use here is

*not *the one they asked for in the first question!

**Second answer to question 2** is P1*P2 = 0.0099, so 11,000 * 0.0099 = 108.9 (109 transactions in 11,000 are expected to be flagged)

Am I even getting close with the order of the probabilities and how i'm multiplying them?

This is still wrong; this time you have correctly calculated what you say, but it is not what they asked. What you have calculated here is P(fraudulent) * P(flagged | fraudulent) = P(fraudulent and flagged). Do you recognize why this is?

But what they asked for is just the number that will be flagged, which is P(flagged) * total number.

You will probably be getting P(flagged) as P(flagged and fraudulent) + P(flagged and not fraudulent).

I know this is complicated, and I have no certainty that I have made it clear. I'd like you to

**ask questions about anything I've said that you aren't sure of**, from the notation for conditional probabilities, to why I think the question means what I say it does.