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?

The probability is 0.6923

Step-by-step explanation:

Let's call R the event that the next day rains, S the event that the next day has sunny weather, R2 the event that the station 2 predicts rain and S1 the event that station 1 predict sunny weather.

The probability that the next day has sunny weather given that station 1 predicts sunny weather for the next day and station 2 predicts rain is calculated as:

P (S/S1∩R2) = P (S∩S1∩R2) / P (S1∩R2)

Where P (S1∩R2) = P (R∩S1∩R2) + P (S∩S1∩R2)

So, the probability P (R∩S1∩R2) that the next day rains, Station 1 predicts sunny weather and Station 2 predicts Rain is calculate as:

P (R∩S1∩R2) = 0.5 * 0.1 * 0.8 = 0.04

Because 0.5 is the probability that the next day rains, 0.1 is the probability that station 1 predicts sunny weather given that it is going to rain and 0.8 is the probability that station 2 predicts rain given that it is going to rain.

At the same way, the probability P (S∩S1∩R2) that the next day has sunny weather, Station 1 predicts sunny weather and Station 2 predicts Rain is calculate as:

P (S∩S1∩R2) = 0.5 * 0.9 * 0.2 = 0.09

Then, the probability P (S1∩R2) that station 1 predicts sunny weather for the next day, whereas station 2 predicts rain is:

P (S1∩R2) = 0.04 + 0.09 = 0.13

Finally, P (S/S1∩R2) is:

P (S/S1∩R2) = 0.09/0.13 = 0.6923