1
$\begingroup$

This question was given to me as a review for an upcoming exam:

If a baseball team wins a game, they have a 40% chance of winning the next game due to getting overconfident. If they lose the previous came, there is a 70% they will win the next game. Assume the first game that there is a 50% of winning. What is the probability the team wins the 4th game?

Is there a way to do this without using total probability?

So far I added all of the 8 scenarios that result in a win on the 4th game to get an answer, but I'm thinking there is a way to do this using matrix multiplication instead of total probability.

$\endgroup$
2
$\begingroup$

Yes, you are right. If state $1$ is win and state $2$ is loss then the transition matrix is $$P=\begin{bmatrix}.4&.6\\.7&.3\end{bmatrix}$$ The probabilities after $4$ games are given by $$\begin{bmatrix}w\\l\end{bmatrix}=P^4\begin{bmatrix}.5\\.5\end{bmatrix}$$ where $w$ and $l$ are the respective probabilities that the fourth game is a win or a loss.

If you look at the matrix multiplication, you will see that this involves exactly the same calculations as the calculations you've already done. You could short-cut it though by squaring $P$ twice. This would cut out one matrix multiplication.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.