A Probability Question (Squash Game)

Question

A squash game consists of rallies problem. 

(Question):-

 

Notes:-

• P(A wins) = ra / (ra+rb), wins a point in a rally
• 0 < ra ≤ 100

 

Suppose player (A) has ability ra = 60 and player (B) has ability rb = 40, and they play a match where the
winner is the first player to win (n) games.

 

What is the smallest value of (n) such that the probability that A wins the match is at least 0.9?

You may answer using simulation, theory, or a combination of both.

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What I got so far that the chance of A beats B is 60% which is 60^n = 0.9 at least ?? I'm sure it's not correct, but please if you got the answer, could you explain it to me? 

 

 

Answer 1

 I'm not sure if there is another way to do this other than trial and error.

 

I started with a best out of 9 series.

 

So A has to win 5 games....This looks like

 

first 5                .6*.6*.6*.6*.6 = .07776

5 out of 6           5C1* .6*.6*.6*.6*.4      *.6 = .15552

5 out of 7           6C2* .6*.6*.6*.6*.4*.4   *.6 = .186624

5 out of 8           7C3 .6*.6*.6*.6*.4*.4*.4    *.6 =  .1741824

5 out of 9           8C4 .6*.6*.6*.6*.4*.4*.4*.4   *.6 = .13934592

 

So this adds to 73.3% which is far off from 90%.

 

I tried the best out of 15, first to win 8. and the probability A wins is only 78.69%

 

This sounds like a computer thing, but I don't have one, so I'm doing by what they call brute force!!!

 

Let's try best out of 29 or the first to 15.

 

win first 15 = .6^15 = .00047

15 of 16     = 15C1*.6^14*.4^1*.6 = .0028

15 of 17     = 16C2*.6^14*.4^2*.6 = .009

15 of 18     = 17C3*.6^14*.4^3*.6 = .0205

15 of 19     = 18C4*.6^14*,4^4*.6 = .0368

15 of 20     = 19C5*.6^14*.4^5*.6 = .056

15 of 21     = 20C6*.6^14*.4^6*.6 = .0746

15 of 22     = 21C7*.6^14*.4^7*.6 = .0896

15 of 23     = 22C8*.6^14*.4^8*.6 = .0985

15 of 24     = 23C9*.6^14*.4^9*.6 = .1007

15 of 25     = 24C10*.6^14*.4^10*.6 = .0967

15 of 26     = 25C11*.6^14*.4^11*.6 = .0879

15 of 27     = 26C12*.6^14*.4^12*.6 = .0762

15 of 28     = 27C13*.6^14*.4^13*.6 = .0633

15 of 29     = 28C14*.6^14*.4^14*.6 = .0506

 

And unfortunately, this only adds up to 86.32%.... so it's higher than the best out of 29....I'll leave the rest to you.

 

Hope this helped.