Question

John is either happy or pouting (a simple soul, our John). If he is happy one day, he is happy the next day four times out of five. If he is pouting one day, the chances that he will also pout the next day are one time out of three. Over the long term, what are the chances that John is happy on any given day?

Answer 1

Answer:

0.7692

Step-by-step explanation:

John is in either if two states: Happy or Sad.

P(Happy)=1/5

P(Sad)=4/5

Let the first column represent the Happy state and the second column represent the Sad Sate.

The transition matrix is:

[TeX]T=\left(\begin{array}{ccc}\frac{4}{5} & \frac{1}{5}\\\frac{2}{3} & \frac{1}{3} \end{array} \right) [/TeX]

We want to determine the equilibrium

matrix p=(x y) by solving pT=p.

[TeX]\left(\begin{array}{cc}x&y \end{array} \right) \left(\begin{array}{ccc}\frac{4}{5} & \frac{1}{5}\\\frac{2}{3} & \frac{1}{3} \end{array} \right)=\left(\begin{array}{cc}x&y \end{array} \right) [/TeX]

gives us the two equations:

[TeX]\frac{4}{5}x+\frac{2}{3}y=x [/TeX]

[TeX]\frac{1}{5}x+\frac{1}{3}y=y [/TeX]

Add -x to the first equation and -y to the second equation.

[TeX]-\frac{1}{5}x+\frac{2}{3}y=0 [/TeX]

[TeX]\frac{1}{5}x-\frac{2}{3}y=0 [/TeX]

The two equations are equivalent so we discard one and replace it with:

x+y=1

[TeX]\frac{1}{5}x-\frac{2}{3}y=0 [/TeX]

This gives: x=0.7692, y=0.2308

Since x=Happy State, the probability that John will be happy in the long run is 0.7692