Question

甲、乙兩支籃球隊(duì)進(jìn)行比賽,已知每一場甲隊(duì)獲勝的概率為0.6,乙隊(duì)獲勝的概率為0.4,每場比賽均要分出勝負(fù).比賽時采用三場兩勝制,即先取得兩場勝利的球隊(duì)勝出.

(1)求甲隊(duì)以二比一獲勝的概率;

(2)求乙隊(duì)獲勝的概率;

(3)(理)若比賽采用五場三勝制,試問甲獲勝的概率是增大還是減小,請說明理由.

Answer 1

解：(1)甲隊(duì)以二比一獲勝,即前兩場中甲勝1場,第三場甲獲勝,其概率為P₁=×0.6×0.4×0.6=0.288.

(2)乙隊(duì)以2∶0獲勝的概率為P₂′=0.4×0.4=0.16;

乙隊(duì)以2∶1獲勝的概率為P₂″=×0.4×0.6×0.4=0.192.

∴乙隊(duì)獲勝的概率為P₂=0.4²+×0.4×0.6×0.4=0.16+0.192=0.352.

(3)(理)若三場兩勝,則甲獲勝的概率

P₃=0.6²+×0.6×0.4×0.6=0.36+0.288=0.648.

或P₃=1-P₂=1-0.352=0.648;若五場三勝,則甲獲勝的概率

P₃′=0.6³+×0.6²×0.4×0.6+×0.6²×0.4²×0.6=0.216+0.259 2+0.207 36=0.682 56.

∵P₃＜P₃′,∴采用五場三勝制,甲獲勝的概率將增大.