Two balls are drawn from an urn containing 2 white, 3 red and 4 black balls one by one without replacement. What is the probability that at least one balls is red?

Let A be the event of not getting a red ball in first draw and B be the event of not getting a red ball in second draw. Then,

Required probability=Probability that at least one ball is red=1–probability that none is  red

= 1 - P(A and B) = 1 -  P (A ∩ B) = 1 - P(A) P(B/A)

Now, P(A) = Probability of not getting a red ball in first draw

P(A) = Probability of getting an other colour (white or black) ball in first draw 

When another colour ball is drawn in first draw there are 5 other colour (white or black) balls and 3 red balls, out of which one other colour ball can be drawn in ⁵C₁  ways.

Substituting these values in (i), we have Required probability