0
$\begingroup$

Team A and Team B are competing in a sports game and the score is currently tied at 10-10. The first team to win by a margin or two will win the tournament. Team A has 65% chance of winning each point and Team B has a 35% chance of winning each point. What is the probability that Team A will win the overall match?

My thoughts: I tried thinking of a Markov chain approach but there are too many states S = {(10-10), (11-10), (12-10), (10-11),....} and would not be practical. If it was just one point I think the probability could be expressed as $p = 0.65 + 0.35\times (1-p)$, but I am not sure how to extend this to two points.

Improve this question
$\endgroup$
5
  • $\begingroup$ Try a different setup for your Markov chain: team A winning a point = +1, team B winning a point is -1. You start at zero, so the states are $\{-2,-1,0,1,2\}$ with $-2$ and $2$ as absorbing states. $\endgroup$
    – jbowman
    Commented Jan 17 at 3:36
  • $\begingroup$ Related? $\endgroup$
    – Dave
    Commented Jan 17 at 4:06
  • 2
    $\begingroup$ Starting from the tied score, consider the future points in sets of 2 rather than each one individually. After playing two points, there are four possible outcomes: (A,A) and A wins the game, (B,B) and B wins the game, (A,B) and the score is tied again, (B,A) and the score is tied again. So now you have a simpler problem to analyze: from the tied score state, you return to the tied score state with probability $x$, go to "A wins" with probability $y^2$, and go to "B wins" with probability $z^2$. Check to be sure that $x+y^2+z^2=1$ before proceeding further. $\endgroup$
    – Dilip Sarwate
    Commented Jan 17 at 14:51
  • $\begingroup$ @jbowman thank you for the suggestion. Would the probability of going from state 0 -> 1 then be 0.65 and 0 -> -1 be 0.35? That way the probability of staying at 0 is 0, but that doesn't take into account both not scoring a point right? $\endgroup$
    – Ria
    Commented Jan 17 at 15:35
  • $\begingroup$ Yes, it would, but @DilipSarwate 's suggestion is better, as it reduces the number of states to 3. $\endgroup$
    – jbowman
    Commented Jan 17 at 16:08

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.