This is more a probabilities problem than a riddle. The riddle is:
I am in a village, where a fisherman lives. The fisherman tells me that there is a 70% possibility that it will rain tomorrow. I know that fisherman's statement's validity is 80%. What is the actual probability that it will rain?
My approach:
We have 2 factors: Let a be the fisherman's prediction and b the fisherman's validity. We are looking for a function $f(a,b)$ with:
- $f,a,b\in[0,1]$
- $f(a,1)=a$
- $f(a,0)=1-a$
- $f(a,b)=f(b,a)$
Am I correct?
- I am sure about (1) and (2).
- Is number (3) correct? Or maybe $f(a,0)$ could be any random number in [0,1]?
- Could (4) be correct? I reached there by thinking that $f(1,b)=b$ and $f(0,b)=1-b$.
How can I continue from here?
Edit: Extra thoughts.
Regarding b-fisherman's validity: What does it mean if b=0?
- One opinion could be that b=0 means that fisherman is always wrong. So f(a,0)=1-a.
- One second opinion would be that fisherman's prediction can be either true or false, without any further clues. So f(a,0)=0.5.
Could in that second case the function be $$f(a,b)=ab+ 0.5(1-b)$$
Moreover, I think that always $f(0.5,b)=0.5$