I have the following question before me: 'This time of the year it rains at some point in the day on about a quarter of the days. On mornings that half the sky is clouded over, it is 10 times more likely that it will rain that day.On days when it is only slightly cloudy in the morning, it is 4 times more likely it is going to rain than it is not going to rain.
Kavya wakes up to find that sky is overcast. What is the probability it is going to rain at some point in the day?'
I am having problem with the language of the question. I am not sure as to what am I actually provided with in this question. If I denote the events of rain and no rain on a particular day by $R$ and $N$ respectively and denote the event of a particular morning being overcast by $O$,does the question say following about the conditional probabilities that $P(R/O)=10P(N/O)$
Am I asked to find the conditional probability $P(R/O)$ simply? I have come up with $P(R/O)=\dfrac{10}{11}$ after noting the fact that $P(R/O)+P(N/O)=1$.Am I interpreting the question correctly? I have written the question verbatim. Please suggest.
Yes! If, given that it is overcast, P(R)= 10P(N). Also, since on any day it must either 'rain' or 'nor rain', P(R)+ P(N)= 1. P(N)= 1- P(R) so P(R)= 10(1- P(R))= 10- 10P(R). 11P(R)= 10. P(R)= 10/11 as you say and P(N)= 1/11.