In Freedonia, 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 $\frac 34$. If it's cloudy on any given day, then the probability that the next day will be cloudy is $\frac 23$.
a. In the long run, what fraction of days are sunny?
b. Given that a consecutive Saturday and Sunday had the same weather in Freedonia, what is the probability that that weather was sunny?
I tried using weighted coins, but that didn't work. Can I get two answers, one for each problem, solution not necessary, as I need to figure out which of my methods leads to the correct answer. Thanks.
I found a congruent problem, but it didn't have answers I could comprehend.
Best Answer
(a) If $p$ is the long-term probability (aka equilibrium point) that it is sunny, then the probability that it is sunny on a following day is also $p$, so: $\Box p + \Box (1-p) = p$
Likewise the probability that it is not sunny on the subsequent day is: $\Box p + \Box (1-p) = (1-p)$.
Fill in the boxes from the given information and solve the simultaneous equation
(b) Use this value as the prior probability that the weather is sunny on Saturday, and construct a Bayesian case for the posterior probability for the given condition.
Let $T$ be the indicator event that it is sunny on Saturday, and $N$ the indicator event that it is sunny on Sunday. We have from the long-term probability: $\mathsf P(T=1)=p, \mathsf P(T=0)=(1-p)$ and also from the given information: $\mathsf P(N=1\mid T=1)=3/4, \mathsf P(N=0\mid T=0)=2/3$.
Find $\mathsf P(T=1 \mid N=T)$ using conditional probability rules.