# Counter intuitive Bayesian theorem

Consider an example as follows.

I am running a mobile app that allows users to apply for a loan on the app.

Say a guy signed in to my app to use his phone to apply for a loan.

Call events:

A = a person has a smartphone

B = default

First of all, I assign P(B)=0.8 for this guy without any information (just be conservative).

Assume that in my country P(A) = 0.5, i.e. only 50% of the population do have a smartphone.

Assume P(A|B) = 1, i.e. when I look to my database, all the guys who did not pay back so far do have a smartphone, that is obvious because users need a smartphone to install my app.

So apply Bayes:

P(B|A) = P(A|B) * P (B) / P(A) = 1 * 0.8 / 0.5 = 1.6

Two problems here indeed:

1) P(B|A) > 1. I know that more than one thread on StackExchange discussed this problem in theory to prove that P(B|A) <= 1 in all cases but could not find why my inference is wrong.

2) Adding one more bit of information, such as “this guy has a smartphone”, according to my Bayesian inference, in fact will increase the probability of default of his case, while in my intuitive inference, it does not bring any information because I know all my customers do have smartphone already. How to explain that?

If instead you just consider users of your app, you might have $$P(B)=0.8$$ and $$P(A)=1$$ and $$P(A \mid B)=1$$. This will now give you $$P(B \mid A)= P(A\mid B) \space P (B) \space / \space P(A) = 1 \times 0.8 / 1 = 0.8$$ and there are no problems there apart from the lack of value in considering $$A$$ since all users of your app have smartphones