A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is?

There are 2 possible cases: first ball drawn is red or is black.
∴P(second ball red | first ball drawn is black)=4/4+8=1/3
∴P(second ball red | first ball drawn is red)=(4+2)/(6+6)=6/12=1/2
Hence, the probability of the second ball being red=P(first ball black)×P(second ball red | first ball black)+P(first ball red)×P(second ball red | first ball red)=6/10×1/3+4/10×1/2=2/10+2/10=4/10=2/5
This is the required probability.