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?

Please help me find it, I’m very tired and I’m stuck p ToT

To solve this problem, we can use conditional probability. We will calculate the probabilities based on the given information.

Let's use the following notation:
- "S" represents a sunny day.
- "R" represents a rainy day.
- "F" represents the announcement of "Fair".
- "B" represents the announcement of "Bad".

(a) The probability we want to calculate is P(S|F), which is the probability that it will be sunny tomorrow given that today's announcement is "Fair".

We know that 75% of all the sunny days were preceded by an announcement of "Fair". So, we start by calculating the probability of having a sunny day preceded by a "Fair" announcement.

P(S and F) = P(S) * P(F|S)
P(S and F) = 0.25 * 0.75
P(S and F) = 0.1875

Next, we need to calculate the probability of receiving a "Fair" announcement.

P(F) = P(S) * P(F|S) + P(R) * P(F|R)
P(F) = 0.25 * 0.75 + 0.75 * (1 - 0.5)
P(F) = 0.1875 + 0.375
P(F) = 0.5625

Now, we can use Bayes' theorem to calculate P(S|F).

P(S|F) = P(S and F) / P(F)
P(S|F) = 0.1875 / 0.5625
P(S|F) = 0.3333

Therefore, if today's announcement is "Fair", the probability of it being sunny tomorrow is approximately 33.33%.

(b) Similarly, we want to calculate P(R|B), which is the probability that it will rain tomorrow given that today's announcement is "Bad".

Based on the given information, we know that 50% of the rainy days were preceded by an announcement of "Bad". So, we start by calculating the probability of having a rainy day preceded by a "Bad" announcement.

P(R and B) = P(R) * P(B|R)
P(R and B) = 0.75 * 0.5
P(R and B) = 0.375

Next, we calculate the probability of receiving a "Bad" announcement.

P(B) = P(S) * P(B|S) + P(R) * P(B|R)
P(B) = 0.25 * (1 - 0.75) + 0.75 * 0.5
P(B) = 0.0625 + 0.375
P(B) = 0.4375

Using Bayes' theorem again, we can calculate P(R|B).

P(R|B) = P(R and B) / P(B)
P(R|B) = 0.375 / 0.4375
P(R|B) ≈ 0.8571

Therefore, if today's announcement is "Bad", the probability of it raining tomorrow is approximately 85.71%.

I hope this explanation helps! If you have any further questions, feel free to ask.