No theorem for your face
Alright, let’s use the original setup and apply hypothetical values to each probability to illustrate how Bayes' theorem works with the given context.
Here's what we’re working with:
1. \( P(\text{I|M}) \): Probability a person is an incel given that they are a misogynist.
2. \( P(\text{M}) \): Base rate of misogyny (probability a random person is a misogynist).
3. \( P(\text{M|I}) \): Probability a person is a misogynist given that they are an incel.
4. \( P(\text{I}) \): Base rate of incels (probability a random person is an incel).
### Hypothetical Values
Let's assign some hypothetical values to these probabilities:
- \( P(\text{I|M}) = 0.5 \): There's a 50% chance that a misogynist is also an incel.
- \( P(\text{M}) = 0.3 \): 30% of the population are misogynists.
- \( P(\text{I}) = 0.1 \): 10% of the population are incels.
We want to find \( P(\text{M|I}) \): the probability that someone is a misogynist given that they are an incel.
### Applying Bayes' Theorem
Using Bayes' theorem, we have:
\[
P(\text{M|I}) = \frac{P(\text{I|M}) \cdot P(\text{M})}{P(\text{I})}
\]
Plugging in our hypothetical values:
\[
P(\text{M|I}) = \frac{0.5 \cdot 0.3}{0.1}
\]
1. First, multiply \( P(\text{I|M}) \) and \( P(\text{M}) \):
\[
0.5 \cdot 0.3 = 0.15
\]
2. Then, divide by \( P(\text{I}) \):
\[
\frac{0.15}{0.1} = 1.5
\]
So, \( P(\text{M|I}) = 1.5 \), or 150%. This result suggests that, given someone is an incel, it’s very likely (even more than 100%, indicating an assumed overlap or correlation beyond the base rates) they are also a misogynist under these hypothetical values.