i have following problem which I am working on:
Say I want to measure something to find out whether it is more likely to belong to class A or class B.
for class A: p = 1
for class B: p = 5
I measure p = 2.9
The measurement is noisy and normally distributed around p with a certain standard deviation.
I want to know the probability that the "something" I measured belongs to class A.
Now I thought about it the following way. I have 2 normal distributions, with $\mu =1$ and $\mu=5$ and the same sigma. Based on my findings in this thread I would calculate the value $z_1=f_1(2.9)$ and $z_2=f_2(2.9)$ where $f_i$ is the probability density function of the first/second normal distribution.
I then get 2 values, let's say $z_1=5$ and $z_2=15$ . This alone shows me that the thing I measured is more likely to belong to class A. But how can I get a specific probability for the membership to class A ?
I do not know at all whether my approach with the PDF is correct, but please point me in another way if it is totally wrong.
You need a priori probabilities. Then you can apply the Bayes' Theorem.
For instance, if the a priori probability of belonging to class A is 0.9, the a posteriori probability would be $$\frac{0.9 \cdot 5}{0.9 \cdot 5 + 0.1 \cdot 15} = 0.75.$$