Question

A bag contains 8 marbles of which 3 are blue and 5 are red. One marble is drawn at random, its colour is noted and the marble is replaced in the bag. A marble is again drawn from the bag and its colour is noted. Find the probability that the marble will be

i. Blue followed by red

ii. Blue and red in any order

iii. Of the same colour

Answer 1

Given:

Bag contains 3 blue and 5 red marbles

It is told that two marbles are drawn with replacement

Let us find the probability of drawing each marble from bag

⇒ P(B) = P(Drawing a Blue Marble)

We need to find the probability that the marbles drawn:

i. Blue followed by red

ii. Blue and red in any order

iii. Of the same colour

⇒ P(S_BR) = P(drawing Blue marble followed by Red)

Since drawing a marble is an independent event, the probabilities multiply each other

⇒ P(S_any) = P(drawing Blue and red marble in any order)

⇒ P(S_any) = P(drawing Blue marble followed by red) + P(drawing Red marble followed by Blue) Since drawing a marble is an independent event, the probabilities multiply each other.

⇒ P(S) = P(drawing two marbles of same colour)

⇒ P(S) = P(drawing black balls from each bag) + (P(drawing white balls from each bag) 

Since drawing a ball is independent for each bag, the probabilities multiply each other.

∴ The required probabilities are \(\cfrac{15}{64},\cfrac{15}{32},\cfrac{17}{32}.\)