Correct : b

Start with 1 Red and 1 Blue ball. At each step, pick one ball at random, then return it along with a new ball of the same colour. After two steps the urn has 4 balls. We want the probability of having exactly 2 Red and 2 Blue.
After Step 1:
Picking Red (probability 1/2) gives urn state (2R, 1B).
Picking Blue (probability 1/2) gives urn state (1R, 2B).
After Step 2 from state (2R, 1B):
Picking Red (probability 2/3) gives (3R, 1B) — unequal.
Picking Blue (probability 1/3) gives (2R, 2B) — equal ✓
After Step 2 from state (1R, 2B):
Picking Red (probability 1/3) gives (2R, 2B) — equal ✓
Picking Blue (probability 2/3) gives (1R, 3B) — unequal.
Total probability:
P(equal) = P(2R,1B) × P(equal | 2R,1B) + P(1R,2B) × P(equal | 1R,2B)
= (1/2 × 1/3) + (1/2 × 1/3)
= 1/6 + 1/6 = 1/3
Correct answer: B — 1/3 ✓