If four caps are picked at random, what is the probability that none is green?

Total number of caps = 2 + 4 + 5 + 1 = 12
Total number of outcomes = n(S) = ¹² C₂ = 66
Favorable number of outcomes = n(E) = ²C₂ = 1
? Required probability = 1/66

Ways of selection of 4 marbles = n(S) = ¹⁵C_{4
Ways of selection of one green marbles = n(E1) = ²C1
Ways of selection of two blue marbles = n(E2)=⁴C2
Ways of selection of one red marble = n(E3) = ⁶C1
? Required probability = (²C1 x ⁴C2 x ⁶C1) / (¹⁵C4) = 24/455}

Ways of selection of two green marbles = ²C₂ = 1
Ways of selection of two yellow marbles = ³C₂ = 3
So, probability (both are green) = 1/¹⁵C₂ ....(I)
Probability (both are yellow) = 3/¹⁵C₂ ....(Ii)

Then, required probability = 1/¹⁵C₂ + 3/¹⁵C₂ = 4/ ¹⁵C₂
= 4/105

Probability that none is blue from four marbles
= ^{15 - 4}C₄ / ¹⁵C₄
= ¹¹C₄ / ¹⁵C₄ = 22/91

So, probability that at least one is blue from four marbles = 1 - 22/91 = 69/91

Ways of selection of two blue marbles = ⁴C₂
Ways of selection of one yellow marble = ³C₁
Ways of selection of three marbles = ¹⁵C₃
So, required probability = (⁴C₂ x ³C₁) / (¹⁵C₃)
= [4!/{2! (4 - 2)!} x 3!/{1! (3 - 1)!}] / [15! / {3! (15 - 3)!}]
= [(4 x 3 x 2 x 1) / (2 x 1 x 2 x 1)] x [(3 x 2 x 1) / (1 x 2 x 1)] / [ (15 x 14 x 13 x 12!) / (3 x 2 x 12!)]
= (6 x 3) / (5 x 7 x 13)
= 18 / 455

Total number of caps = 12
? n(S) = ¹²C₃ = 220
n(E₁) = Out of 4 red caps, number of ways to pick 2 caps = ⁴C₂ = 6
n(E₂) = Out of 5 green caps
Number of ways to pick one cap = ⁵C₁ = 5
P(E) = n(E₁) x n(E₂)/n(S) = (6 x 5)/220 = 3/22

Required probability = 1 - ⁸C₂/ ¹²C₂
= 1 - 28/66 = 38/66 = 19/33

Total number of possible outcomes
= n(S) = ¹²C₂ = (12 x 11)/2 = 66

Total number of favorable events
= n(E) = ⁴C₂
= (4 x 3) / (2 x 1) = 6
? Required probability = 6/66 = 1/11

Total Number of outcomes = ¹⁰C₃ = 120
Number of outcomes not containing red balls = ⁵C₃ = 10
? Probability that at least one is red = 1 - 10/120 = 11/12

Total number of outcomes = ¹⁰c₄ = 210
Favourable number of outcomes = ³c₂ x ²c₂ = 3 x 1 = 3
? Required probability = 3/210 = 1/70