Probability Question - What is the probability that it will rain on the day of Marie's wedding Marie is getting married tomorrow, at an outdoor ceremony in the desert. In recent years, it has rained only $5$ days each year.
Unfortunately, the weatherman has predicted rain for tomorrow. When it actually rains, the weatherman correctly forecasts rain $90\%$ of the time. When it doesn't rain, he incorrectly forecasts rain $10\%$ of the time.
What is the probability that it will rain on the day of Marie's wedding?
 A: Let $F$ represent the event that he forecasted rain will occur.  Let $R$ represent that rain actually occurs.
We are told the following information:
$Pr(F\mid R)=0.9$ (if it rained, the forecaster predicted it correctly 90% of the time)
$Pr(F\mid R^c)=0.1$ (if it didn't rain, the forecaster incorrectly predicted it should have rained 10% of the time)
We are also likely intended to assume that $Pr(R)=\dfrac{5}{365}$ as per the phrase "in recent years it has rained only 5 days each year" or perhaps $Pr(R)=\dfrac{5}{365.25}$ or some other similar number according to how much you wish to account for leap year.  For simplicity's sake, I would suggest using $\dfrac{5}{365}$.
We are tasked with calculating $Pr(R\mid F)$.
Now... with this information we can apply Bayes' theorem:
$$Pr(R\mid F)=\dfrac{Pr(F\mid R)Pr(R)}{Pr(F)}$$
The terms in the numerator were given directly in the problem.  The term in the denominator can be calculated using the law of total probability and conditional probability:
$Pr(F)=Pr(F\cap R)+Pr(F\cap R^c)=Pr(R)Pr(F\mid R)+Pr(R^c)Pr(F\mid R^c)$
Plug in the appropriate numbers and reach a final conclusion.
