Q5. An urn has 6 red and 4 black balls. Three are drawn without replacement. Find the probability that all three are red.
(6/10)(5/9)(4/8) = 120/720 = 1/6.
Q6. P(A) = 0.6, P(B|A) = 0.5 and P(A|B) = 0.75. Find P(B).
P(A ∩ B) = P(A) P(B|A) = 0.3, and P(A ∩ B) = P(B) P(A|B), so P(B) = 0.3/0.75 = 0.4.
Q7. A bag has 3 red and 4 white balls. Find the probability that two drawn balls are both red (i) with replacement; (ii) without replacement.
(i) (3/7)(3/7) = 9/49. (ii) (3/7)(2/6) = 1/7, which is smaller, because the first red ball is not put back.
Q8. In NCERT Example 8, let E be “the first ball is black” and F be “the second ball is black”. Find P(F) and P(E|F), and check that P(F) P(E|F) also gives 3/7.
P(F) = (10/15)(9/14) + (5/15)(10/14) = 3/7 + 5/21 = 2/3, the same as for the first ball. P(E|F) = P(E ∩ F)/P(F) = (3/7)/(2/3) = 9/14. Then P(F) P(E|F) = (2/3)(9/14) = 3/7.
Q9. Cards are drawn one at a time without replacement from a pack of 52. Find the probability that the first ace appears on the third draw.
Two non-aces and then an ace: (48/52)(47/51)(4/50) = 376/5525, about 0.068.