probability

91.
(a) (i)
No. of permutations when the 6 girls and 7 boys queue up randomly
= P(13,13)
= 13!

No. of permutation when Yen and Mandy are at both end
= P(2, 2) x P(11,11)
= 2! x 11!

P(Yen and Mandy are at both ends)
= (2! x 11!)/13!
= 2/(13 x 12)
= 1/78

(a) (ii)
Number of permutations when girls are at both ends
= P(6, 2) x P(11, 11)
= (6!/4!) x 11!
= (6 x 5) x 11!
= 30 x 11!

P(Girls are at both ends)
= (30 x 11!)/13!
= 30/(13 x 12)
= 5/(13 x 2)
= 5/26


(b)
No

Girl is denoted as G, and boy is denoted by B.

The number of B is one more than that of G. When any boys are not next to eachother, the only case is that :
BGBGBGBGBGBGB

However, the number of G is one less than that of B. When any girls are notnext to each other, there are a few cases :
BGBGBGBGBGBGB, GBBGBGBGBGBGB, GBGBBGBGBGBGB ......

Obviously, P(Any girls are not next to each other) > P(Any boys are not nextto each other)


(c)
No

"Boys are not next to each other" : BGBGBGBGBGBGB
P(Boys are not next to each other)
= P(7,7) x P(6,6)/13!
= 7!6!/13! ≠ 1/2

P(Boys are next to each other)
= 1 - P(Boys are not next to each other)
≠ 1/2

Hence, P(Boys are not next to each other) ≠ P(Boys are next to each other)