if from each of the three boxes containing 3 white and 1 black, 2 white and 2 black and 1 white and 3 black balls, one ball is drawn at random, then the probability that 2 white and 1 black ball will be drawn is?

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

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

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

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 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 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

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

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

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- 1/6) = (3/4) x (5/6)
= 5/8