Porbability question: what percentage of free throws....

[leilsilver]

For this question:

If I made my lst free throw, there is an 80 % chance I will make my next one. On the other hand, if I missed my last shot, there is a 60 % chance I will miss my next one as well. What percentage of my shots do I make, in the long run?

The way my teacher taught me to do this was that we made a punnett square/matrix thing and we raised it to the hundredth like. Is that how I would do this?

 

[pka]

You want what is known as a Markov matrix:
\(\displaystyle \L
\left( {\begin{array}{lr}
{.8} & {.2} \\
{.6} & {.4} \\
\end{array}} \right).\)

This Markov change reaches steady state by the fifth power:
\(\displaystyle \L
\left( {\begin{array}{lr}
{.8} & {.2} \\
{.6} & {.4} \\
\end{array}} \right)^{5} =
\left( {\begin{array}{lr}
{.75} & {.25} \\
{.75} & {.25} \\
\end{array}} \right).\)