When A predicts raining, the chance of raining is 60%. When B predicts raining, the chance of raining is also 60%. If A and B both predict to rain (assuming they did the prediction independently), what is the chance of raining?
My understanding of the problem is as follows:
Let $X, Y, Z$ respectively denote the events of A predicts raining, B predicts raining, and it does rain. Then we have: $$P(Z|X)=P(Z|Y)=.6,\,P(XY)=P(X)P(Y).$$ We need to find out $P(Z|X,Y)$.
I could not figure out how to calculate the it. Maybe I can only get a range of it?
Thanks in advance!
Let $X,Y,Z$ be the events that $A$ predicts rain, $B$ rpredicts rain, it rains. We are given that $X,Y$ are independant and $P(Z|X)=P(Z|Y)=0.6$.
It could be that $P(X)=P(Y)=0.6$ and $Z=X\cap Y$. In that case $P(Z|A,B)=1$.
It could also be that $P(X)=P(Y)=0.4$ and $Z=\overline{X\cap Y}$. In that case $P(Z|A,B)=0$.
And anything inbetween is possible.