How to determine long-run probability on a calculator and manually?
For example:
Ben plays a tennis match every day.
If he wins on one particular day, the probability that he wins the next day is 0.8
If he loses one day, the probability that he loses the next day is 0.6
The long-run probability that Ben wins a match is:
How to calculate this "by hand" and "by a calculator"
What I have gathered using my knowledge on conditional probability:
- $P(w_2|w_1) = 0.8$
- $P(w_2'|w_1')= 0.6$
Trying to find P(Ben wins a match in the long run) = ?
I know the answer, but am unsure how to get there.
Best Answer
Let $W_t$ be win today, $W_y$ be win yesterday, $L_t$ be lose today and $L_y$ be lose yesterday.
We have $P(W_t)=P(W_t|W_y)\cdot P(W_y)+P(W_t|L_Y)\cdot P(L_Y)$.
From the given information, this becomes $P(W_t)=.8P(W_y)+.4P(L_y)$.
If the probabilities are to be stable (long-term), we should have $P(W_y)=P(W_t)$ and $P(L_y)=P(L_t)$.
Thus $P(W_t)=.8P(W_t)+.4P(L_t)$. Also $P(W_t)+P(L_t)=1$, which should be enough information to solve the problem.
Another approach would be to set up a Markov transition matrix, and find the stable probability vector.