Question

There is a bag filled with 4 blue and 5 red marbles.

A marble is taken at random from the bag, the colour is noted and then it is replaced
Another marble is taken at random.
What is the probability of getting 2 of the same colour?

Answer 1

41/81

Explanation:

There are a total of 4 blue marbles, 5 red marbles giving 9 total number of marbles

Let P(B₁) =Probability of getting a blue marble on the first draw

Let P(B₂) =Probability of getting a blue marble on the second draw

Let P(R₁) =Probability of getting a red marble on the first draw
Let P(B₂) =Probability of getting a red marble on the second draw

P(B₁) = Number of blue marbles ÷ total number of marbles = 4/9

Since the marble is replaced before the second draw, the total number of blue marbles and total number of both marbles is the same as for the first draw

P(B₂) = P(B₁) = 4/9 also

Probability of getting 2 blue marbles on 2 consecutive draws= P(B₁B₂)
= 4/9 x 4/9 = 16/81

P(R₁) = Number of red marbles ÷ total number of marbles = 5/9

P(getting a red marble on the second draw) = P(R₁R₂) = 5/9

Probability of getting 2 red marbles on 2 consecutive draws = P(R₁R₂)
= 5/9 x 5/9 = 25/81

Since we are asked the probability of getting 2 of the same color we can rephrase the same question as

P(two of same color) = P(B₁B₂ ∪ R₁R₂) = P(B₁B₂) + P(R₁R₂) = 16/81 + 25/81 = 41/81