Given:

Bag contains 4 white, 7 black and 5 red balls

It is told that four balls are drawn with replacement.

We need to find the probability that at least two balls drawn are white.

Let us find the individual probabilities to draw a ball

⇒ P(W) = P(drawing a white ball)

⇒

⇒

⇒ P(O) = P(drawing balls other than white)

⇒

⇒

The possible cases of drawing balls are as shown below:

i. Two whites and two others – 6 cases to draw

ii. Three whites and 1 others – 4 cases to draw

iii. Four whites – 1 case to draw

Since drawing balls are independent events so, the probabilities multiply each other.

⇒

⇒

⇒

⇒

∴ The required probability is .