In the old scorekeeping system for badminton, you score a point when you win a rally as the
server. If you win a rally as the receiver, the score remains unchanged, but you get to serve
and thus the opportunity to score. Suppose Ann wins a rally against Bob with probability p,
regardless of who serves (this is a reasonable assumption in badminton, unlike tennis, where
the server often has the advantage). Assume rallies are won independently of each other. Ann
is currently the server, and let X denote the number of rallies until the next point is won.
• Find a difference equation that characterizes the PMF of X.
The probability that Ann wins the next point is p, and Bob winning is (1-p)^2. Then there is p*(1-p) if Ann loses then wins. Then I think I am supposed to use the law of total probability next, giving:
1*p + 1*(1-p)^2 + p*(1-p)*[something]
Now do I just take what I solve from that and make it into a PMF?
server. If you win a rally as the receiver, the score remains unchanged, but you get to serve
and thus the opportunity to score. Suppose Ann wins a rally against Bob with probability p,
regardless of who serves (this is a reasonable assumption in badminton, unlike tennis, where
the server often has the advantage). Assume rallies are won independently of each other. Ann
is currently the server, and let X denote the number of rallies until the next point is won.
• Find a difference equation that characterizes the PMF of X.
The probability that Ann wins the next point is p, and Bob winning is (1-p)^2. Then there is p*(1-p) if Ann loses then wins. Then I think I am supposed to use the law of total probability next, giving:
1*p + 1*(1-p)^2 + p*(1-p)*[something]
Now do I just take what I solve from that and make it into a PMF?