maths probability Q3+Q4

Q3 :a) P(John will win)= P(John scores 4 and Mary scores 2 or 3)
+ P(John scores 6 and Mary scores 2 or 3 or 4 or 5)
+ P(John scores 8 and Mary scores 2 or 3 or 4 or 5 or 6 or 7)
+ P(John scores 10 and Mary scores 2 or 3 or 4 or 5 or 6 or 7 or 8 or 9)
+ P(John scores 12 and Mary scores 2 or 3 or 4 or 5 or 6 or 7 or 8 or 9 or 11)= (1/6) (1/36)(1+2)
+ (1/6) (1/36)(1+2+3+4)
+ (1/6) (1/36)(1+2+3+4+5+6)
+ (1/6) (1/36)(1+2+3+4+5+6+5+4)
+ (1/6) (1/36)(1+2+3+4+5+6+5+4+3+2)= (1/6) (1/36) (99)= 11/24
b) P(Mary will win)= P(John scores 2 and Mary scores 3 or 4 or 5 or 6 or 7 or 8 or 9 or 11 or 12)
+ P(John scores 4 and Mary scores 5 or 6 or 7 or 8 or 9 or 10 or 11 or 12 )
+ P(John scores 6 and Mary scores 7 or 8 or 9 or 10 or 11 or 12)
+ P(John scores 8 and Mary scores 9 or 10 or 11 or 12)
+ P(John scores 10 and Mary scores 11 or 12)Since the number of ways of :
11 or 12 = 2 or 3 ,
9 or 10 or 11 or 12 = 2 or 3 or 4 or 5
7 or 8 or 9 or 10 or 11 or 12 = 2 or 3 or 4 or 5 or 6 or 7
etc .By symmetry ,P(Mary will win) = P(John will win) = 11/24
c)P(There is a tie game)= 1 - 11/24 - 11/24= 1/12 Q4 :a) It need to move upwards 3 times and right 1 times.
(3+1)! / (3! 1!) = 4 ways.P(E) = 4 * (1/2)^(3+1) = 1/4
b) It need to move upwards 2 times and right 3 times.
(2+3)! / (2! 3!) = 10 ways.P(E) = 10 * (1/2)^(2+3) = 5/16
c) 4(1/4) + 3(5/16) = 1 + 15/16

2011-01-24 16:33:28 補充：
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P(John scores 12 and Mary scores 2 or 3 or 4 or 5 or 6 or 7 or 8 or 9 or 10 or 11)