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.

Short answer

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

Step-by-step solution

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.

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

localid="1648052534806" role="math" $x_3=P^3{x}_0={\left[\begin{array}{cc}0.7& 0.3\\ 0.3& 0.7\end{array}\right]}^3.\left[\begin{array}{c}0.5\\ 0.5\end{array}\right]$

Now,

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

That is

localid="1648051526652" $x_3=\left[\begin{array}{c}0.5\\ 0.5\end{array}\right]$

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