Two similar boxes B_i (i = 1, 2) contain (i + 1) red and (5 – i – 1) black balls. One box is chosen at random and two balls are drawn randomly. What is the probability that both the balls are of different colours?

Concept:

•  → choosing or selecting r elements from n elements.

Calculation:

Clearly, B1​ has 2 red and 3 black balls whereas B2​ has 3 red and 2 black balls.

P(B1) = P(B2) = 1/2 

We need probability of getting both the balls of different colours

i.e., Required probability, P = P(B1)P(B)P(R) + P(B2)P(B)P(R)

Here for bag 1, we need to select 1 black and 1 red out of 3 black and 2 red balls respectively 

Similarly, for bag 2

P = P(B1)P(B)P(R) + P(B2)P(B)P(R) 

⇒  

(∵ Choosing one black and red out of 3 and 2 respectively from the B₁ of 5 balls

Similarly, from B₂ of 5 balls choosing one black and red out of 2 and 3 respectively.)

     (∵​ and )​

⇒ 3/10 + 3/10

⇒ 6/10

⇒ 3/5

Hence, option (4) is correct.​