If two marbles are drawn at random, what is the probability that both are red?

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

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

Total number of caps = 12
Total number of result = n(S) = ¹²C₄ = 495
Out of 5 caps, number of ways to not pick a green cap = n(E₁) = ⁵C₀ = 1
and out of 7 caps, number of ways to pic 4 caps = n(E) = ⁷C₄ = 35
? Required probability = (1 x 35)/495 = 7/99

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}

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

n(S) = Number of ways of drawing 2 cards out of 52
⁵²C₂ = (52 x 51) / (2 x 1) = 1326
n(E) = Number of ways of drawing 2 cards out of 4
= ⁴C₂ = (4 x 3)/2 = 6
? P(E) = 6/1326 = 1/221

There are 13 hearts and 3 more kings
? P(heart or a king) = (13 + 3)/52 = 4/13
? P (neither a heart nor a king) = 1 - 4/13 = 9/13

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