I'm attempting a solution to the following problem, copied verbatim from my problem set. (Apparently the problem comes from this book.)
Every evening, two weather stations issue weather forecast for the next day. The weather forecasts are independent. On average, the weather forecast of station 1 is correct 90% of the cases, irrespective of weather type; this percentage is 80% for station 2. Suppose that, without any forecast, the belief is that the next day rains or has sunny weather each with a 50% chance. On a given day, station 1 predicts sunny weather for the next day, whereas station 2 predicts rain. What is the probability that the weather forecast of station 1 will be correct?
I understand the problem is supposed to test my comprehension of conditional probability. But I'm having trouble formalizing the information. Here is my attempt.
Let
- $S$ denote the event "tomorrow it will, in fact, be sunny"
- $R=S^c$ denote the event "tomorrow it will, in fact, rain"
Then the second bolded sentence means the unconditional probabilities are $P(S)=P(R)=0.5$.
What does the first bolded sentence mean? I guess it means
$$P(S\,|\,\text{Station 1 predicts sun})=0.9$$ $$P(R\,|\,\text{Station 1 predicts rain})=0.9$$ $$P(S\,|\,\text{Station 2 predicts sun})=0.8$$ $$P(R\,|\,\text{Station 2 predicts rain})=0.8$$
Then, I think the question is asking for the value of
$$P(S\,|\,\text{Station 1 predicts sun and Station 2 predicts rain})$$
If this is correct, I know that by the definition of conditional probablity the result is
$$\frac{P(S\,\text{and Station 1 predicts sun and Station 2 predicts rain})}{P(\text{Station 1 predicts sun and Station 2 predicts rain})}$$
But I don't know how to proceed from here. Even if I use "the weather forecasts are independent" to conclude that the denominator factors, I don't know the probability that Station 1 predicts sun; indeed it doesn't seem like that event is stochastic, so it doesn't have a probability.
So I think my setup must be wrong.