Question

甲、乙兩隊(duì)進(jìn)行一場(chǎng)排球比賽，根據(jù)以往經(jīng)驗(yàn)，單局比賽甲隊(duì)勝乙隊(duì)的概率為0.6.本場(chǎng)比賽采用五局三勝制，即先勝三局的隊(duì)獲勝，比賽結(jié)束.設(shè)各局比賽相互間沒(méi)有影響.?

(理)令ξ為本場(chǎng)比賽的局?jǐn)?shù),求ξ的概率分布和數(shù)學(xué)期望.(精確到0.000 1)?

(文)求(1)前三局比賽甲隊(duì)領(lǐng)先的概率；?

(2)求本場(chǎng)比賽乙隊(duì)以3∶2取勝的概率.(精確到0.001)

Answer 1

(理)解析:單局比賽甲隊(duì)勝乙隊(duì)的概率為0.6,乙隊(duì)勝甲隊(duì)的概率為1-0.6=0.4.?

比賽3局結(jié)束有兩種情況:甲隊(duì)勝3局或乙隊(duì)勝3局,因而?

P(ξ=3)=0.6³+0.4³=0.28.?

比賽4局結(jié)束有兩種情況:前3局中甲隊(duì)勝2局,第4局甲隊(duì)勝或前3局中乙隊(duì)勝2局,第4局乙隊(duì)勝.因而?

P(ξ=4)=×0.6²×0.4×0.6+×0.4²×0.6×0.4=0.374 4.?

比賽5局結(jié)束有兩種情況:前4局中甲隊(duì)勝2局\,乙隊(duì)勝2局,第五局甲勝或乙勝,因而?

P(ξ=5)= ×0.6²×0.4²×0.6+×0.6²×0.4²×0.4=0.345 6.?

所以ξ的概率分布為

ξ	3	4	5
P	0.28	0.374 4	0.345 6

ξ的期望Eξ=3×P(ξ=3)+4×P(ξ=4)+5×P(ξ=5)=3×0.28+4×0.374 4+5×0.345 6=?4.065 6.??

(文)解析:單局比賽甲隊(duì)勝乙隊(duì)的概率為0.6,乙隊(duì)勝甲隊(duì)的概率為1-0.6=0.4.?

(1)記“甲隊(duì)勝三局”為事件A,“甲隊(duì)勝兩局”為事件B,則P(A)=0.6³=0.216,?

P(B)= ×0.6²×0.4=0.432.?

所以,前三局比賽甲隊(duì)領(lǐng)先的概率為P(A)+P(B)=0.648.?

(2)若本場(chǎng)比賽乙隊(duì)以3∶2取勝,則前四局雙方應(yīng)以2∶2戰(zhàn)平,且第五局乙隊(duì)勝.?

所以,所求事件的概率為×0.4²×0.6²×0.4≈0.138.