Question

There are ten balls in a bag. Three balls are black and seven balls are white. a One ball is chosen at random and then replaced. Then a second ball is chosen at random. Here are two events. D: the first ball is black S: the second ball is black Are D and S independent? Show your working. ​

Answer 1

Given:

• Total balls = 10
• Black balls = 3
• White balls = 7

⇒ P(black ball) = 3/10 = 0.3

⇒ P(white ball) = 7/10 = 0.7

If one ball is chosen at random and replaced, the probabilities do not change between the first and second ball.

Let event D = first ball is black

Let event S = second ball is black

P(D) = 0.3

P(S) = 0.3 (since the first ball is replaced)

⇒ P(D ∩ S) = P(D) × P(S)

= 0.3 × 0.3

= 0.09

For Independent Events we know that: P(A ∩ B) = P(A) × P(B)

Therefore, as P(D ∩ S) = P(D) × P(S) the events are independent.