Question

(2) In some sports (such as volleyball when I was in high school - scoring rules have changed) you can only score a point when it is your serve, and losing when it is your serve does not give up a point but only gives up the serve to your opponent. Suppose your probability of scoring on your serve is p (and of losing the serve is 1 − p) while your opponent’s probability of scoring on their serve is q (so the probability of you regaining the serve is 1 − q). Find a formula for the probability S that you are the first to score the next point, assuming you currently have the serve. (This takes evaluating an infinite series.) While on the previous part, I have walked you through the process of figuring out what infinite series to evaluate and asked you explicit questions to answer, you should go through the same steps (answering similar questions) and write them out here. As before, get actual numbers for various values of p and q and comment on whether these are what you would expect.

(2) In some sports (such as volleyball when I was in high school - scoring rules have changed) you can only score a point when it is your serve, and losing when it is your serve does not give up a point but only gives up the serve to your opponent. Suppose your probability of scoring on your serve is p (and of losing the serve is 1-p) while your opponent's probability of scoring on their serve is q (so the probability of you regaining the serve is 1-9). Find a formula for the probability S that you are the first to score the next point, assuming you currently have the serve. (This takes evaluating an infinite series.) While on the previous part, I have walked you through the process of figuring out what infinite series to evaluate and asked you explicit questions to answer, you should go through the same steps (answering similar questions) and write them out here. As before, get actual numbers for various values of p and q and comment on whether these are what you would expect.

Answer 1

Suppose your probability of scoring on your serve is p (and of losing the serve is 1 − p) while your opponent’s probability of scoring on their serve is q (so the probability of you regaining the serve is 1 − q).

Assuming it is your serve:

1. As it is your serve, Probability you will score a point without losing serve = p

2. Probability you will score a point after losing 1 serve = (1-p)(1-q)*p

3. Probability you will score a point after losing 1 serve = (1-p)(1-q)(1-p)(1-q)*p

and so on until infinite serves.

We can see that the probability of score a point first after losing n serve is given by

$P(x=n)=(1-p)^{n-1}(1-q)^{n-1}p^n$

As, the above term is a geometric series with,

a = p

r = (1-p)(1-q)

The sum of an infinite geomtric series

$S_{\inf} =\frac{a}{1-r}$

Therefore the overall probability that you will score a point first, is

1-(1-P)(1-9)

Using above formula, we can calculate the probability for different values of p and q as in table below

p	q	Probability of scoring first
0.2	0.2	=0.2/(1-(1-0.2)(1-0.2)) =0.555556
0.4	0.2	0.769231
0.6	0.2	0.882353
0.8	0.2	0.952381
0.2	0.4	0.384615
0.4	0.4	0.625
0.6	0.4	0.789474
0.8	0.4	0.909091
0.2	0.6	0.294118
0.4	0.6	0.526316
0.6	0.6	0.714286
0.8	0.6	0.869565
0.2	0.8	0.238095
0.4	0.8	0.454545
0.6	0.8	0.652174
0.8	0.8	0.833333

We notice that when value of q remains constant and p increases, the probability of you scoring the first point increases.

Similarly, when value of q increases with p being constant the probability of you scoring the first point decreases.

But there are values of p and q where probability is not expected. For example, when p= 0.2, and q =0.2 or 0.4, the probability of first score is 0.555 and 0.385 respectively. This shows that, despite the probability of winning a point(p=0.2) being very low or equal compared to q, the overall probability of scoring first point remains high for the person taking the first serve.