Math EX.[F5]

2013-02-16 10:59 pm

Help me to slove the following questions

1.The 6 volumes of a set of comic are arranged at random on a bookshelf.
Find the probabilitics that the volumes are arranged
a) exactly in the order of the volume numbers
b) with the 5th and the 6th volumes together
c) with the 1st and the 6th volumes at the two ends

2. A quiz contains 5 true-or-false questions. If Rachel answers all the questions
by wild guessing, what are the probabilities that she answers
a) exactly 4 questions correctly
b) at least 4 questions correctly

回答 (1)

賣女孩的火柴

2013-02-17 4:52 am

✔ 最佳答案

1.
(a)
Number of arrangements without restriction
= 6P6
= 6!

Number of arrangements in exact order
= 1

The required probability
= 1/6!
= 1/720

(If reversed order, i.e. 6,5,4,3,2,1, is alsoaccepted,)
(the answer will be 2/720 = 1/360)

(b)
Put the 5th and the 6th volumes together as a pair.
The internal permutation of the pair of these 2 volumes = 2P2
Arrange this pair of the 2 volumes and other 4 volumes (5P5).

Number of the required arrangements
= 2P2 x 5P5
= 2! x 5!

The required probability
= 2! x 5! / 6!
= 1/3

(c)
Put the 1st and the 6th volumes at the two ends (2P2).
Arrange the rest 4 volumes (4P4).

Number of the required arrangements
= 2P2 x 4P4
= 2! x 4!

The required probability
= 2! x 4! / 6!
= 1/15

2.
(a)
P(1 correct) = 1/2
P(1 incorrect) = 1/2

P(exactly 4 correct)
= P(4 correct and 1 incorrect)
= 5C4 x (1/2)⁴ x(1/2)
= (5!/4!1!) x (1/2)⁵
= 5/32

(b)
P(at least 4 correct)
= P(4 correct and 1 incorrect) + P(5 correct)
= (5/32) + (1/2)⁵
= (5/32) + (1/32)
= 3/16

參考: 賣女孩的火柴

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