Say there is a $20$ percent chance of rain on Saturday, $30$ percent chance of rain on Sunday. If the two events are independent of each other, then the chance of it raining on at least one of the days is$$.2 + .3 – (.2)(.3) = .44.$$However, if I'm told that raining on Saturday makes it more likely to rain on Sunday, will the probability go up or down that it will rain on at least one day? I'm not sure how to answer this question, hence asking here.
Raining and independence on the weekend
algebra-precalculusexpected valueindependenceprobabilityrandom variables
Best Answer
Let A="it rains on Saturday", B="it rains on Sunday".
You are asked P(A or B)=P(A)+P(B)- P(A and B)
The last term can be written as P(A and B)=P(B|A) P(A) If A and B are independent, then P(B|A)=P(B). However, if rain on Saturday makes rain on Sunday more likely, then P(B|A)>P(B), and hence P(A or B) will decrease, as
P(A or B)=P(A)(1-P(B|A))+P(B)
and P(A) and P(B) remained constant.