Answer:
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
is 
, so any range such that 
will give two complete periods.