A tub contains 15 black balls and 10 white balls out of which 4 black balls and 2 white balls are defective. If a person takes two balls at random, what is the probability that either both are black balls or both are good balls?

A tub contains 15 black balls and 10 white balls out of which 4 black balls and 2 white balls are defective. If a person takes two balls at random, what is the probability that either both are black balls or both are good balls? Correct Answer 221/300

Given:

A tub contains 15 black balls and 10 white balls out of which 4 black balls and 2 white balls are defective

Concept used:

The classical definition of probability.

Formula used:

Probability = Favorable Outcome/Total Outcome.

P(E) = n(E)/n(S)

Calculation:

Number of ways in which two balls can be taken from 25 balls = ²⁵C₂ 

Number of ways in which two black balls can be taken from 15 black balls = ¹⁵C₂ 

P(both are black balls) = ¹⁵C₂/²⁵C₂

Number of ways in which two balls can be taken from 19 good balls = ¹⁹C₂ 

Total good balls = (15 + 10) - (4 + 2) = 19

P(both are good) = ¹⁹C₂/²⁵C₂

There are 11 good black balls. Number of ways in which two black balls can be taken from these 11 good black balls = ¹¹C₂

P(both are black balls and both are good) = ¹¹C₂/²⁵C₂

P(both are black balls or both are good) = P(both are black balls) + P(both are good) – P(both are black balls and both are good)

(¹⁵C₂/²⁵C₂) + (¹⁹C₂/²⁵C₂) − (¹¹C₂/²⁵C₂)

⇒ (¹⁵C₂ + ¹⁹C₂ – ¹¹C₂)/²⁵C₂

⇒ /