RAS question
A bag has 5 red and 3 blue balls. Two balls are drawn without replacement. The probability that both are red is:
Correct answer: (D) 5/14.
When two balls are drawn without replacement from a bag containing 5 red and 3 blue balls, the probability that both balls are red is 5/14.
Explanation
NCERT's probability chapter treats such questions through conditional probability: once an event has occurred, the sample space for the next event is reduced to the outcomes still possible. Here the first draw is red with probability 5/8 because 5 of the 8 balls are red. Since the draw is without replacement, a red first draw leaves 4 red balls among 7 total balls, so the conditional probability of the second ball also being red is 4/7. Multiplying the two stages gives 5/8 x 4/7 = 20/56 = 5/14. The same result comes from counting equally likely pairs: C(5,2)/C(8,2) = 10/28 = 5/14.
Why the other options are wrong
- (A) 3/8 does not account for the ordered two-stage calculation, because after one red ball is removed the second probability is 4/7, not a fixed fraction that leads to 3/8.
- (B) 5/8 is only the probability that the first ball is red; it ignores the required condition that the second ball must also be red.
- (C) 25/64 would come from multiplying 5/8 by 5/8, which treats the second draw as if the first ball were replaced, contrary to the question.
Concept
This tests conditional probability for drawing without replacement. It recurs in RAS reasoning because aspirants must adjust the sample space after each draw instead of applying a mechanical one-draw fraction.