mathematical expectation

heng87

a student guesses at random each and every one of the 50 objective questions in a PMR examination.If each question has 4 possible answers,what is the variance of X,where X is the number of questions answered correctly?

答案是9.375.我不是很明白题目的意识。请解释题目的意识和如何算到答案


two badminton players,Alan and Brown,are to play in a match for a wager.The first person to win sets in a row,or two out of three sets wins the match.If Alan wins the match,Brown pays him b ringgits per played,whereas if Brown wins the match,Alan pays Brown a ringgit per set played.The probability of Alan winning the first set is 2/3.However,Alan is a veteran player who tires easily to the extent that subsequently his probability of winning any set is one half that of winning the previous set.Find the ratio a:b which makes the wager "fair" to both players.
答案是a:b=13:33 .我不是很明白题目的意识。请解释题目的意识和如何找到答案

數學神童

for the first question
n=50,p=1/4,q=3/4
variance=npq

peaceboy

b的答案确定没错？

heng87

回复 3# peaceboy
是的

peaceboy

回复  peaceboy
是的
heng87 发表于 24-9-2010 10:08 PM

之前没看到是Alan pays Brown a ringgit per set played

E(A win) = (2/3)(1/3)(2b) + (2/3)(2/3)(1/6)(3b) + (1/3)(1/3)(1/6)(3b)
            =4a/9+6b/27+3b/54
            =39b/54

E(B win) = (1/3)(2/3)(2a) + (1/3)(1/3)(5/6)(3a)+(2/3)(2/3)(5/6)(3a)
            = 4a/9+15a/54+30a/27
            =99a/54

when the game is fair , 39b/54=99a/54
39b=99a
a/b = 39/99
     =13/33
a:b=13:33

heng87

回复 5# peaceboy
E(A win) = (2/3)(1/3)(2b) + (2/3)(2/3)(1/6)(3b) + (1/3)(1/3)(1/6)(3b)

    (2b),(3b),(3b)是怎么样得到的
E(B win) = (1/3)(2/3)(2a) + (1/3)(1/3)(5/6)(3a)+(2/3)(2/3)(5/6)(3a)
    (2a),(3a),(3a)是怎么样得到的

请解释，谢谢

peaceboy

回复  peaceboy
E(A win) = (2/3)(1/3)(2b) + (2/3)(2/3)(1/6)(3b) + (1/3)(1/3)(1/6)(3b)

    (2b),( ...
heng87 发表于 25-9-2010 05:32 PM

    (2/3)(1/3)<<<<玩两盘就赢，所以获得2b
(2/3)(2/3)(1/6)<<<玩三盘赢，所以获得3b
(1/3)(1/3)(1/6)<<<玩三盘赢，所以获得3b


(1/3)(2/3) <<玩两盘就赢，所以获得2a
(1/3)(1/3)(5/6)<<<玩三盘赢，所以获得3a
(2/3)(2/3)(5/6)<<<玩三盘赢，所以获得3a