Statistics / Probability - Why do these two functions look this way?

Posted on 2015-10-16 06:35:00 in Math by JEFFY

Hello, I have a two functions -- f and g -- and I'm trying to figure out why their graphs look the way they are. A little background info: 



1) Players A and B have a probability of p and 1-p to score a point, respectively. Players with at least four points and a two point lead on the other win the game. 



2) Players with at least 6 won games and a 2 game lead on their opponent win the set. 



Based on p, I calculated the probability that A wins a game, and the probability A wins a set. 



Binomial(x,n,p) is the binomial probability that a scores x out of n points, with probability p. 



f(p) = P(A wins game) = 

binomial(3,6,p)*p*(1-p)*2 * [ (p^2) / ((p^2) + (1-p)^2)) ] 

+ p^4 

+ p*binomialf(3,4,p) 

+ p*binomial(3,5,p) 

+ p*p*binomial(3,6,p)) 



let k = P(A wins a game) 



g(k) = P(A wins set) = 

binomial(5,10,k)*k*(1-k)*2 * [ (k^2) / ((k^2) + (1-k)**2) ] 

+ k^*6 

+ k*binomial(5,6,k) 

+ k*binomial(5,7,k) 

+ ... 

+ ... 

+ k*k*bin.pmf(5,10,k) ) 



---------------------------------------... 

Now my question: The graphs for the two functions are in the picture (below?). g(k) is much steeper than f(p). Can anyone tell me why this is? I know it has something to do with g being a function of f. Any help or hints are greatly appreciated! Thank you

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