Maths and sta question 求救 20點

[匿名]

2011-06-16 3:28 am

2.John and Peter play Chinese chess. Theprobabilities that john wins and draws in a game are p and 0.5 respectively,where 0<p<1. The winner gets 2 marks, the loser gets 0 mark and eachplayer gets 1 marks when the game ends with a draw. (a) Find the probabilities , interm of p, that Peter wins in a game.(b) John and Peter play two gamesfirst. The one who gets a higher total score is the winner .If they get anequal total score , then they will play another two games. The process willcontinue until there is winner. Find the probability in term of p , that there is a winner afterplaying two games.(c) If ther probabilities thatthere is a winner af ter playing two games is greater than 0.63, find the rangeof possible values of p.(d) Can the probability that thereis a winner after playing two games be greater than 0.8?Explain your ans .

回答 (1)

andrew

2011-06-16 9:11 am

✔ 最佳答案

2.(a)
P(Peters wins in a game)
= 1 - P(John wins in a game) - P(Draw in a game)
= 1 - p - 0.5
= 0.5 - p

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2.(b)
P(John wins after 2 games)
= P(John wins in 2 games) + P([John wins in 1 game] and [draw in 1 game])
= p² + 2C1xpx0.5
= p² + p

P(Peter wins after 2 games)
= P(Peter wins in 2 games) + P([Peter wins in 1 game] and [draw in 1 game])
= (0.5 - p)² + 2C1x(0.5-p)x0.5
= 0.25 - p + p² + 0.5 - p
= p² - 2p + 0.75

P(there is a winer after 2 games)
= P(John wins after 2 games) + P(Peter wins after 2 games)
= p² + p + p² - 2p + 0.75
= 2p² - p + 0.75

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2.(c)
2p² - p + 0.75 > 0.63
2p² - p + 0.12 > 0
200p² - 100p + 12 > 0
50p² - 25p + 3 > 0
(5p - 1)(10p - 3) > 0
p< 1/5 or p > 3/10
p < 0.2 or p > 0.3

But 0 < p ≤ 0.5
Hence, range of p: 0 < p < 0.2 or 0.3 < p ≤ 0.5

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2.(d)
P(John wins a game) + P(draw in a game) ≤ 1
p + 0.5 ≤ 1
p ≤ 0.5
2p ≤ 1
2p - 1 ≤ 0
p(2p - 1) ≤ 0
2p² - p ≤ 0
2p² - p + 0.75 ≤ 0.75
P(there is a winer after 2 games) ≤ 0.75
Hence, the probability cannot be0.8.

參考: andrew

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