Question

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.

Answer 1

| Solution : Suppose your probablity of Scoxing on your cenice is a land of losing the serve is 1-22 whaite your opponants probablity of scoring on their serve is also the probablity of you Tregoining the serve is 1-2) *Assuming it is your serveis As it is your serve, probablity you will score a point without losing serve ah 21 probablity you will score a point after losing serve (1-P) C-94 p. اله طحنهمههعمللتدخلها للطحطمتعلق Hosing serve a CL-P) CL-9) (1-9) and so on unti infinite serves 0 he can see that the probablity of score a pointed barst losing a serve is given by CamScanner

probablity of scoring first جلد مجله ماعدهم کیاکہ = (ر2، و 234و 0 : 26 بیسک23 ههههههههه (GS23 .210 ، ه ، ه ا كاوه . له. هبل ده . . 62. 6 ۔ : ها: يا هلا و . . . ووو 18 م العموم.6 هههههههههههه 1. :ملحف ظها و دهه && 2 ولكنه | 6 . 6 | 6 ها كك مع .6 معه مساب رم يوقعه كهعههه 6. 63244 6 ،8 6.6 33 333 )، ه CamScanner

we notice that when value of a remains constant and increales, the probablity of you scoring that first point increases, Similarly when value of q increas with a being constant the probablity of you scoring the first point decreases. But they are values of p and a where probablity is not expected. Roy exclample wher عمل هفته معه لمعدهم د والطحطه Rirst score is .05551280 0.35 gespectively This shows Hiat deposite the probablity of محبيها وجعي "ومعطلحمع لخطتهعهقصص eauad probablity of scoring Piast point Tremains high for the person taking the Rixst serveen co CamScanner
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