Tow persons A and B appear in an interview for two vacancies. If the probabilities of their selections are 1/3 and 1/6 respectively, then the probability that none of them is selected, is

Total lottery tickets = 10000
Total prize in the lottery = 10
? Probability getting a prize
= 10/10000 = 1/1000

Now, probability of not getting a prize
= 1 - probability of getting a prize
= 1 - (1/1000)
= 999/1000

Required probability = P(A) x P(B)
= (1 - 1/4) x (1 - 1/3) = 3/4 x 2/3 = 1/2

Total number of ways
= n(s) = ⁶c₂ = 15
Favorable number of ways
= n(E) = ³C₁ x ³C₁= 9
? Required probability = 9/15 = 3/5

Total balls in the box = 5
Second red ball can be drawn in two ways
Case I. First ball is white and second ball is red.
Its probability = 3/5.2/4 = 6/20 = 3/10
Case II: First ball is red and second ball is red
Its probability = 2/5.1/4 = 2/20 = 1/10
Hence, reqd. probability
= 3/10 + 1/10 = 4/10 = 2/5

You can have
W, W, B or W, B, W or B, W, W

Reqd. probability
= 3/4. 2/4. 3/4 + 3/4. 2/4. 1/4 + 1/4. 2/4. 1/4
= 9/32 + 3/32 + 1/32 = 13/32

Total ways = ¹³c₂
Favourable number ways of selecting men only = ⁸c₂
? Probability of selecting no woman
= ⁸c₂ / ¹³c₂
= 14/39
? Probability of selecting at least one woman
= 1 - (14 / 39)
= 25 / 39

Total events = n (s) = 2⁵ = 32
n(E) of getting heads = 1
p(E) = 1/32
? n(E) = 1 - p(E) = 1 - 1/32 = 31/32

Let P(B) = x
Given, P(A?B) = 0.8 and P(A) = 0.3
? P(A) + P(B) - P(A?B) = 0.8
? P(A) + P(B) - P(A) P(B) = 0.8 {?A and B are independent}
? 0.3 + x - 0.3x = 0.8
? 0.7x = 0.5
? x = 5/7

Let the number of green marble = x
Then, Probability of getting a green marble
= ^xc₁ / ²⁴c₁ = 2/3
? x/24 = 2/3
? x = 16

Here, total balls are 14.
? Required probability = (³c₁ + ⁷c₁) / ¹⁴c₁
= (3 + 7)/14
= 10/14
= 5/7