# Trouble understanding Bayes Theorem

I was watching a video on YouTube and i am not sure if the given solution is correct. Can someone confirm? • The final + in the denominator should be x. – BruceET Oct 15 '18 at 7:50
• There is no such a thing as "naive Bayes theorem", there are naive Bayes algorithm and Bayes theorem. – Tim Oct 15 '18 at 7:52

## 2 Answers

Seems correct, except for the typo noted in my Comment. Let $$D$$ indicate 'has disease' and $$T$$ indicate 'tests positive'.

Bayes' Theorem states the following (denoting intersection of events as 'multiplication'):

$$P(D|T) = \frac{P(DT)}{P(T)} = \frac{P(D)P(T|D)}{P(DT)+P(D^cT)} = \frac{P(D)P(T|D)}{P(D)P(T|D)+P(D^c)P(T|D^c)}.$$

Sometimes, $$P(T) = P(D)P(T|D)+P(D^c)P(T|D^c)$$ is called the Law of Total Probability.

You are given that $$P(D) = 0.01,\, P(T|D) = P(T^c|D^c) = 0.99.$$ Then by the Complement Rule, $$P(D^c) = 0.99,\, P(T|D^c) = 0.01.$$ Plug in these numbers to get the answer claimed.

To understand these probabilities, notice that $$P(DT), P(D|T),$$ and $$P(T|D)$$ all contain the same events, but they refer to three different populations: the first to the population of everyone who may take the test, the second to the population of those who tested positive, and the third to the population of those who have the STD.

Note: You can find more detailed discussions of this kind of situation in several of the links under 'Related' in the right margin. Also, you may want to look at Wikipedia articles on "Bayes' Theorem" and "screening test."

Yes even without consulting the equations it is possible to work it out from the information. See below. • This is a nice presentation, but I think you have some typos? The If Test Neg And Have AIDS group tests negative, but are included in the Sum people who test pos sum. And it should only be (1,000,000 - 10,000) * 1% = 9900 people who test positive and don't have AIDS. So the 0.5 answer is correct, but it should be 9900 / (9900 + 9900), not 10k / 20k. And (less important) you have the number of people with AIDS labeled as a percent. – Gregor Oct 15 '18 at 14:42
• Quibble: Question referred to STD and you say AIDS. Because of the seriousness of HIV disease, screening tests typically have much higher sensitivity $P(T|D)$ and (especially) specificity $P(T^c|D^c)$ than in ordinary medical screening tests. – BruceET Oct 15 '18 at 15:41
• @BruceET Good catch. – Curtis White Oct 16 '18 at 15:41