JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Probability

WhatsApp Email

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is

\frac{m}{n}

, where

gcd (m, n) = 1

, then

m + n

is equal to :

(A)14✓ Correct
(B)4
(C)11
(D)13

Answer: (A)

Solution

Bag has 4 white and 6 black balls, total 10. Two balls are drawn one by one without replacement. We want

P (1st black ∣ 2nd black) = \frac{P ( 1st black and 2nd black )}{P ( 2nd black )} .

P (both black) = \frac{6}{10} \times \frac{5}{9} = \frac{30}{90} = \frac{1}{3} .

P (2nd black)

— by the law of total probability, splitting on the color of the first ball:

P (2nd black) = P (1st white) P (2nd black ∣ 1st white) + P (1st black) P (2nd black ∣ 1st black) = \frac{4}{10} \times \frac{6}{9} + \frac{6}{10} \times \frac{5}{9} .

= \frac{24}{90} + \frac{30}{90} = \frac{54}{90} = \frac{3}{5} .

P (1st black ∣ 2nd black) = \frac{\frac{6}{10} \times \frac{5}{9}}{\frac{4}{10} \times \frac{6}{9} + \frac{6}{10} \times \frac{5}{9}} = \frac{30/90}{54/90} = \frac{30}{54} = \frac{5}{9} .

m = 5, n = 9

, and

m + n = 14

JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Single correct option

Practise more: all JEE Main PYQs on Probability →

JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Probability

Solution

More JEE Main questions on Probability