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Today I wanted to help someone with a probability exercise, it was like this:

If it's sunny today, it's gonna be sunny tomorrow with probability $0.7$. If it rains today, it will rain tomorrow with probability $0.8$. (It can only either be sunny or rain).

Now the questions was to calculate the expected amount of sunny days in a year ($360$ days). Is that possible by knowing only the conditional probabilities? I couldn't figure it out..

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Let it be that $p$ denotes the probability that some fixed day will appear to be a sunny day.

Then the probability that the day after this day is sunny equals: $$p\times0.7+\left(1-p\right)\times0.2$$ so that:$$p=p\times0.7+\left(1-p\right)\times0.2$$ and consequently $p=0.4$.

The expectation of the number of sunny days among $360$ is then $360\times0.4=144$.

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