In a city, every day is either cloudy or sunny (not both). If it's sunny on any given day, then the probability that the next day will be sunny is 3/4. If it's cloudy on any given day, then the probability that the next day will be cloudy is 2/3.
a) In the long run, what fraction of days are sunny?
b) Given that a consecutive Saturday and Sunday had the same weather in the city, what is the probability that that weather was sunny?
To calculate the fraction of sunny days in the long run, we need to consider the probabilities of having consecutive sunny or cloudy days.
a) Let's denote the fraction of sunny days as "p." We can set up the following equation:
p = (3/4) * p + (1 - p) * (2/3)
Here, (3/4) * p represents the probability of a sunny day followed by another sunny day, and (1 - p) * (2/3) represents the probability of a cloudy day followed by a sunny day.
Now, let's solve for p:
p = (3/4) * p + (2/3) - (2/3) * p
Multiplying both sides of the equation by 12 to eliminate the fractions, we get:
12p = 9p + 8 - 8p
Simplifying further:
12p - 9p + 8p = 8
p = 8/11
Therefore, in the long run, approximately 8/11 (or about 0.727) of the days will be sunny.
b) To find the probability that the weather was sunny given that a consecutive Saturday and Sunday had the same weather, we can use Bayes' theorem.
Let's define the following events:
- S: Weather is sunny.
- C: Weather is cloudy.
- W: Weather on Saturday and Sunday is the same.
We need to calculate P(S | W), which represents the probability that the weather is sunny given that Saturday and Sunday had the same weather.
Using Bayes' theorem, we have:
P(S | W) = (P(W | S) * P(S)) / P(W)
Here,
- P(W | S) represents the probability that Saturday and Sunday would have the same weather if the weather is sunny.
- P(S) represents the probability of having a sunny day.
- P(W) represents the probability of having the weather on Saturday and Sunday the same.
From the given information, we know that if it's sunny, the probability that the next day will be sunny is 3/4. Therefore, P(W | S) = (3/4) * (3/4) = 9/16.
We previously calculated P(S) as 8/11.
To calculate P(W), we need to consider the two possibilities: sunny-sunny and cloudy-cloudy. The probability of getting two sunny days in a row is (8/11) * (3/4) = 24/44. The probability of getting two cloudy days in a row is (3/11) * (2/3) = 6/33.
So, P(W) = (24/44) + (6/33) = 12/22 + 2/11 = 38/66.
Finally, substituting the values into Bayes' theorem:
P(S | W) = (9/16) * (8/11) / (38/66)
Simplifying the fraction:
P(S | W) = (9/16) * (8/11) * (66/38)
P(S | W) = 9/19
Therefore, the probability that the weather was sunny given that a consecutive Saturday and Sunday had the same weather is 9/19.