Intereting Posts

Showing a ring where $ax = b$ has a solution for all non-zero $a, b$ is a division ring
Is it wrong to use Binary Vector data in Cosine Similarity?
Limit of $n/\ln(n)$ without L'Hôpital's rule
Can non-constant functions have the IVP and have local extremum everywhere?
Can multiplication be defined in terms of divisibility?
Contour integration to compute $\int_0^\infty \frac{\sin ax}{e^{2\pi x}-1}\,\mathrm dx$
The distance function on a metric space
On the hessian matrix and relative minima
Is any divergence-free curl-free vector field necessarily constant?
Smallest positive integer that gives remainder 5 when divided by 6, remainder 2 when divided by 11, and remainder 31 when divided by 35?
Why do people use “it is easy to prove”?
Three-dimensional simple Lie algebras over the rationals
Proving that a sequence has $a_n = a_{n+2}$ for $n$ sufficiently large.
Show that $\frac{(3^{77}-1)}{2}$ is odd and composite
$\mathbb{Z}^{3}/\langle(b,6,0)\rangle$ according to the structure theorem for finitely generated abelian groups

I’ve been studying probability to develop a more intuitive sense of calculating probabilities as a medical practitioner. One example that came up in discussing the importance of *prior probabilities* was HIV testing. In the example the book **The Laws of Medicine by Siddhartha Mukherjee** gives, he gives the problem of calculating the probability of someone having HIV given that they have a positive ELISA test. Here’s what we know:

The probability of HIV being present in the population: $P(A) = \frac{1}{1000}$

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- Show independence of four random variables as combination of four other independent variables
- Intuitive/heuristic explanation of Polya's urn

The probability of a false positive (Getting a positive test ($B$) result given that the person doesn’t have HIV ($A’$)): $P(B|A’) = \frac{1}{1000}$

We want to find the probability that someone has HIV given that they have a positive test result: $P(A|B)$?

My first thought is that we don’t have enough information to solve this. Here’s the math to back this up:

$P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{P(A)P(B|A)}{P(B)}$

From the information we’re given, there’s no way of finding $P(B|A)$. So based on the information given in the book, there’s no way of calculating $P(A|B)$. Is this right? If not, I’d like to find a pathway towards calculating $P(B|A)$ using the information I’m provided. The big thing here for me is to understand intuitively when some probability isn’t computable and if it is computable, is there a simple/intuitive way to do it. Thanks!

- Estimating the transition matrix given the stationary distribution
- How do I calculate the probability distribution of the percentage of a binary random variable?
- Why the principal components correspond to the eigenvalues?
- Probability that $xy = yx$ for random elements in a finite group
- How many distinct n-letter “words” can be formed from a set of k letters where some of the letters are repeated?
- Past coin tosses affect the latest one if you know about them?
- Sum of two independent geometric random variables
- Expectation of random variables ratio

Rough intuition: imagine a population of 1000 people that perfectly reflects the statistics. Then you’ll get one positive test from the one infected person (assuming there are negligible false negatives) and one positive test from the 999 people who are HIV free. So the probability of a positive test indicating real illness is 1/2.

I think this method (sometimes called “natural frequencies”) offers more insight than Bayes’ Theorem on conditional probabilities.

Read http://opinionator.blogs.nytimes.com/2010/04/25/chances-are/ to see how often medical practitioners get this wrong.

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