Finding True Positive Probability

Question

Problem

Find the true positive of infected covid-19 patients.

It's been 3 years since I've done any probability so I'm very rusty with this stuff. Any help is much appreciated.

	Total Patients	Tested Positive
Sample size	2000
Infected	1000	990
Not Infected	1000	20

My Attempt

Definitions: True Positive means $P(test^+|covid^+)$ according to the problem statement given.

Here's my approach. Let $c^+$ mean covid positive and $t^+$ mean tests positive. I use the conditional probability formula: $$P(t^+|c^+) = \frac{P(t^+ \text{ and } c^+)}{P(c^+)}$$.

So I proceed by finding the and first.

$$P(t^+ \text{ and } c^+) = \frac{990}{1000}$$

Then I find $P(c^+) = \frac{1000}{2000}$. Then I plug these into the conditional probability formula:

$$P(t^+ | c^+) = \frac{\frac{990}{1000}}{\frac{1000}{2000}} = \frac{99}{50} = 1.98$$

This can't be right since it's greater than 1. I just can't figure out what I'm doing wrong.

No, you computed $P[t^+ \cap c^+]$ incorrectly, the denominator is the entire collection, so $P[t^+ \cap c^+] = {990 \over 2000}$. — copper.hat, Jul 30 '22 at 19:44
That is how you compute the probability of any event, The number of times the event happens divided by the total number. — copper.hat, Jul 30 '22 at 20:27
See https://math.stackexchange.com/questions/2279851/applied-probability-bayes-theorem/2279888#2279888 — Ethan Bolker, Jul 30 '22 at 21:16

true blue anil · Answer 1 · 2022-07-30T21:10:15.477

1

Actually, things will be much simpler if you consider numbers who tested positive against numbers actually diseased amongst those tested

A definition here says "The fraction (percentage) of subjects with a given disease correctly identified as such. Also known as true positive rate or sensitivity."

Thus Pr = n(tested positive in those tested)/n(diseased in those tested) $= \frac{990}{1000}$

PS

Actually, a less technical (and hence generally clearer) is applying Bayes' theorem through what is known as a contingency table. When a part of it has been provided, as it has been here, you can just read off values easily.

edited Jul 30 '22 at 21:10

answered Jul 30 '22 at 20:33

true blue anil

41,295

Yeah, the ultimate goal of the assignment is to use Bayes' Theorem, I just thought that these probabilites would be calculated with conditional ones first. Then use Bayes for subsequent questions that the problem asks. I'm just a bit confused on which to use (Bayes or conditional) and how to calculate the and probability. – amGz Jul 30 '22 at 21:29

Finding True Positive Probability

Problem

My Attempt

1 Answers1