甲乙两人进行乒乓球比赛,甲单局获胜的概率是p(p>1/2).若采取5局3胜制.设获胜的概率为q,问当p为何值时,p-q取得最大值

Question

甲乙两人进行乒乓球比赛,甲单局获胜的概率是p(p>1/2).若采取5局3胜制.设获胜的概率为q,问当p为何值时,p-q取得最大值

Answer 1

甲获胜的概率有以下情况
三场结束： p^3
四场结束： (C3 2)p^2*(1-p)*p
五场结束： (C4 2)p^2*(1-p)^2*p
q=p^3+3p^3(1-p)+6p^3(1-p)^2
p-q=p-p^3-3p^3(1-p)-6p^3(1-p)^2
=p(1-p)(1+p)-3p^3(1-p)-6p^3(1-p)(1-p)
=(p^2+p-3p^3-6p^3+6p^4)(1-p)
=(p+p^2-9p^3+6p^4)(1-p)
(p-q)'=(1+2p-27p^2+24p^3)(1-p)-(p+p^2-9p^3+6p^4)
=1+p-28p^2+33p^3-6p^4-p-2p^2+27p^3-24p^4
=1-30p^2+60p^3-30p^4
=1-30p^2(p-1)^2
导数=0时
p约等於76%
p约76%时,p-q最大
p约为1-76%时,p-q最小,但是这里p>1/2不考虑了