Q. 9.5 - A First Course in Probability

Question

Consider Example 2a. If there is a 50–50 chance of rain today, compute the probability that it will rain 3 days from now if α = .7 and β = .3. 

Step-by-Step Solution

Answer

The probability that it will rain $3$ day from now is $\frac{1}{2}$.

Step 1: Given Information

We have given that there is 50–50 chance of rain today.

We need to find the probability that it will rain three days from now.

Step 2: Simplify

Given a transition matrix 

$P=\left[\begin{array}{cc}0.7& 0.3\\ 0.3& 0.7\end{array}\right]$

and the beginning distribution ( the distribution of $X_0$) is

$x_0=\left[\begin{array}{c}0.5\\ 0.5\end{array}\right]$

To find the distribution of $X_3$we have

$x_3=P^{3 } x_{0 }={\left[\begin{array}{cc}0.7& 0.3\\ 0.3& 0.7\end{array}\right]}^3.\left[\begin{array}{c}0.5\\ 0.5\end{array}\right]$

Now,

$P^3=\left[\begin{array}{cc}0.532& 0.468\\ 0.468& 0.532\end{array}\right]$

That is 

$x_3=\left[\begin{array}{c}0.5\\ 0.5\end{array}\right]$

So, that means there is the probability of  $\frac{1}{2}$ that it will rain on the third day.