Two beads are drawn at random, one after the other, with replacement, from a box containing 5 red and 7 white…

Analyze the question: Two beads are drawn with replacement, so the two draws are independent. We want the probability that both beads are the same colour, meaning both red or both white.

Concept: For independent events, P(A and B) = P(A) × P(B). For "either/or" cases that cannot happen together, we add the probabilities.

Solution:

Total beads = 5 + 7 = 12

• P(red) = 5/12
• P(white) = 7/12

P(both red) = (5/12) × (5/12) = 25/144

P(both white) = (7/12) × (7/12) = 49/144

P(same colour) = 25/144 + 49/144 = 74/144 = 37/72

Answer: The probability is 37/72, which is option C.

Check: P(different colours) = 2 × (5/12)(7/12) = 70/144. Then 74/144 + 70/144 = 144/144 = 1 ✓

Tip: "Same colour" means adding up the probabilities of each colour pairing (RR + WW). Option D (49/144) is a trap, because it is only the "both white" case. Always check that you have included every case. A quick test is to confirm that "same" and "different" add up to 1.