急! F5 Probability 05q8

33.
(a)
P(4C gives an incorrect answer)
= (1 - 0.6) * (1 - 0.8) * (1 - 0.75)
= 0.02

P(4C gives a correct answer)
= 1 - 0.02
= 0.98

(b)
P(4C wins the contest)
= P(4C gets 90 marks or 100 marks)
= P(4C gives 9 correct answers) + P(4C gives 10 correct answers)
= C(10,9)*(0.98)^9*(0.02) + (0.98)^10
= 0.9671


37.
(a)
P(B) = 0.4, then P(B') = 1 - 0.4 = 0.6

P(A|B') = P(A∩B')/P(B')
0.3 = P(A∩B')/0.6
P(A∩B') = 0.3 * 0.6
P(A∩B') = 0.18

(b)
P(A) = a, then P(A') = 1 - a

P(B|A') = P(A'∩B)/P(A')
0.5 = P(A'∩B)/(1 - a)
P(A'∩B) = 0.5(1 - a)

(c)
P(A'∪B)
= P(A') + P(B) - P(A'∩B)
= (1 - a) + 0.4 - 0.5(1 - a)
= 1 - a + 0.4 - 0.5 + 0.5a
= 0.9 - 0.5a

(A'∪B)' = A∩B'
P(A'∪B)' = 0.18
1 - (0.9 - 0.5a) = 0.18
1 - 0.9 + 0.5a = 0.18
0.5a = 0.08
a = 0.16


38.
(a)
Total number of lucky stars
= 7 + 4x + 3x
= 7x + 7

P(2 yellow lucky stars)
= P(1st lucky star is yellow) x P(2nd lucky star is yellow)
= [7/(7x + 7)] * [6/(7x + 6)]
= [1/(x + 1)] * [6/(7x + 6)]
= 6/(7x² + 13x + 6)

P(2 yellow lucky stars) = 1/10
6/(7x² + 13x + 6) = 1/10
7x² + 13x + 6 = 60
Hence, 7x² + 13x - 54 = 0

(b)
From (a) :
7x² + 13x - 54 = 0
(x - 2)(7x + 27) = 0
x = 2 or x = -27/7(rejected)

Number of blue lucky stars
= 4*2
= 8

Number of red lucky stars
= 3*2
= 6

(c)
Method 1 :
P(1 yellow lucky star)
= 7/(7x + 7)
= 1/(x + 1)
= 1/(2 + 1)
= 1/3

P([another lucky star is yellow] | [1 yellow lucky star])
= P(2 yellow lucky stars) / P(1 yellow lucky star)
= (1/10) / (1/3)
= 3/10

Method 2 :
Total number of lucky stars
= 7x + 7
= 21

When 1 yellow lucky star is drawn, 20 lucky stars are left in the jar, and 6 ofthem are yellow.
P([another lucky star is yellow] | [1 yellow lucky star])
= 6/20
= 3/10