In the state of Campana, during the Fall season, it rains 75% of the time and it is sunny 25% of the time (these figures are based on historical data). The local weather service every day announces either “Fair” or “Bad”. In the past, 75% of all the sunny days were preceded (the day before) by an announcement of “Fair” by local weather service and 50% of the rainy days were preceded by an announcement of “Bad”.
(a) Suppose that today the local weather service announces “Fair”. What is the probability that it will be sunny tomorrow?
(b) Suppose that today the local weather service announces “Bad”. What is the probability that it will rain tomorrow?
To solve this problem, we can use Bayes' Theorem, which is a formula used to calculate conditional probability. Bayes' Theorem states:
P(A|B) = (P(B|A) * P(A)) / P(B)
where P(A|B) is the probability of event A given event B, P(B|A) is the probability of event B given event A, P(A) is the probability of event A, and P(B) is the probability of event B.
(a) Let's calculate the probability that it will be sunny tomorrow given that today the local weather service announces "Fair."
P(Sunny|Fair) = (P(Fair|Sunny) * P(Sunny)) / P(Fair)
From the given information, we know that P(Fair|Sunny) = 0.75 (75% of the sunny days were preceded by an announcement of "Fair") and P(Sunny) = 0.25 (25% of the time it is sunny).
We need to calculate P(Fair), which can be obtained by adding the probabilities of all possible scenarios: P(Fair|Sunny) * P(Sunny) + P(Fair|Rainy) * P(Rainy).
P(Fair|Rainy) is not given in the question, so we need to calculate it using complementary probabilities:
P(Fair|Rainy) = 1 - P(Bad|Rainy) = 1 - 0.50 = 0.50 (50% of the rainy days were preceded by an announcement of "Bad").
Since P(Rainy) = 0.75 (75% of the time it is rainy), we can calculate P(Fair):
P(Fair) = P(Fair|Sunny) * P(Sunny) + P(Fair|Rainy) * P(Rainy)
= 0.75 * 0.25 + 0.50 * 0.75
= 0.1875 + 0.375
= 0.5625
Now we can substitute these values into the equation to find P(Sunny|Fair):
P(Sunny|Fair) = (0.75 * 0.25) / 0.5625
= 0.1875 / 0.5625
= 0.3333
Therefore, the probability that it will be sunny tomorrow given that today the local weather service announces "Fair" is 0.3333 (or 33.33%).
(b) Now let's calculate the probability that it will rain tomorrow given that today the local weather service announces "Bad."
P(Rainy|Bad) = (P(Bad|Rainy) * P(Rainy)) / P(Bad)
From the given information, we know that P(Bad|Rainy) = 0.50 (50% of the rainy days were preceded by an announcement of "Bad") and P(Rainy) = 0.75 (75% of the time it is rainy).
We need to calculate P(Bad), which can be obtained by adding the probabilities of all possible scenarios: P(Bad|Sunny) * P(Sunny) + P(Bad|Rainy) * P(Rainy).
P(Bad|Sunny) is not given in the question, so we need to calculate it using complementary probabilities:
P(Bad|Sunny) = 1 - P(Fair|Sunny) = 1 - 0.75 = 0.25 (25% of the sunny days were preceded by an announcement of "Bad").
Since P(Sunny) = 0.25 (25% of the time it is sunny), we can calculate P(Bad):
P(Bad) = P(Bad|Sunny) * P(Sunny) + P(Bad|Rainy) * P(Rainy)
= 0.25 * 0.25 + 0.50 * 0.75
= 0.0625 + 0.375
= 0.4375
Now we can substitute these values into the equation to find P(Rainy|Bad):
P(Rainy|Bad) = (0.50 * 0.75) / 0.4375
= 0.375 / 0.4375
= 0.8571
Therefore, the probability that it will rain tomorrow given that today the local weather service announces "Bad" is 0.8571 (or 85.71%).