Question 6: The IRS is looking for ways to identify fraud....

Question 6: The IRS is looking for ways to identify fraud. The agency is aware that many criminals claim to have unprofitable small businesses as a way to reduce their tax obligations. Let A represent the event that a filer claims to have an unprofitable small business. Let B represent the event that a filer is committing criminal fraud. Assume $P(B) = 0.001$, $P(A | B) = 0.85$, $P(A^c | B^c) = 0.99$. Given that a filer has claimed to have an unprofitable small business, what is the probability he is committing criminal fraud? [Hint: For any arbitrary events X and Y, $P(X|Y^c) = 1 - P(X|Y^c)$ holds.]

Transcript

00:01 Hey there, welcome to numerate.

00:04 So we are looking at a current tax year, where there's a 5 % of fraudulent tax returns.

00:11 We can express this as our probability here for our binomial distribution, knowing that our sample size here that we're looking at are the 250 tax returns.

00:23 So knowing this, our binomial equation is the probability of x equals our binomial coefficient.

00:31 Times the probability raised to x times q raised to the n minus x so for part a here what we have here is the probability that at least 15 are fraudulent returns so knowing this here we have the probability of x being greater than or equal to 15 so this here will basically be equivalent to the probability of 15 plus the probability of 16 and we are going to keep them going upwards here until we reach the sample size of 250.

01:15 Therefore all of these probabilities in between are going to be plugged into our equation above and it's summed all together given us a probability of around 0 .271 all right? so let's try part b over here.

01:40 So what we're looking for for part b is that there's only a 90 % chance that irs will uncover at least 15.

01:50 So therefore we have at least 15 fraudulent returns is going to be 1 minus the 0 .9.

02:00 So therefore we have the probability of x being less than or equal to.

02:08 So let's see, actually, sorry, greater than or equal to 15, which equals what we have here as the probability of x is less than 15.

02:25 So if we expand it out like above here, what we'll get here is going to be, let me see here, sorry, this should be equals 15.

02:42 And we're going to go upwards, 16 again, all the way to.

02:48 Our number of trials of 250.

02:56 Therefore, here, let's see what we get for our probability...

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