Question

Suppose a sunny day is followed by a sunny day with probability 0.7, and a cloudy day is followed by a sunny day with probability 0.5.
a. if it is sunny today, what is the probability that it will be sunny 3 days later?

Answer 1

Establish a transition matrix:

$T = \left[\begin{array}{ccc}0.7&0.3\\0.5&0.5\end{array}\right]$

The possible path are:

sunny -> cloudy -> cloudy -> sunny

-> sunny -> sunny

-> sunny -> cloudy -> sunny

-> sunny -> sunny

Find and add all the paths using the transition matrix T:

0.3*0.5*0.5 + 0.3*0.5*0.7 + 0.7*0.3*0.5 + 0.7*0.7*0.7 = 0.628