A bag contains 5 red and 7 green balls. Two balls are drawn at random, one after the other, without replacement. The probability that both balls are of the same colour is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{31}{66}\)
“Same colour” means both red or both green. These two cases cannot happen together, so their probabilities are added. Without replacement, the bag has one ball fewer for the second draw.
\( \displaystyle P(\text{both red}) = \frac{5}{12}\times\frac{4}{11} = \frac{20}{132} \)
\( \displaystyle P(\text{both green}) = \frac{7}{12}\times\frac{6}{11} = \frac{42}{132} \)
Adding:
\( \displaystyle P(\text{same colour}) = \frac{62}{132} \)
\( \displaystyle = \frac{31}{66} \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{35}{66}\) — Incorrect. This is \(1 - \frac{31}{66}\), the probability that the two balls are of different colours.
Option (3) \(\displaystyle\frac{37}{72}\) — Incorrect. This is the answer with replacement: \(\frac{25}{144} + \frac{49}{144} = \frac{74}{144}\).
Option (4) \(\displaystyle\frac{5}{33}\) — Incorrect. This is \(\frac{20}{132}\), the probability of both red only.