Question

There is a bag filled with 3 blue, 4 red and 5 green 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 the same colour?

Answer 1

Let A = {The second marble has the same colour with first taken marble}

$H_i =$ {First taken marble has color i} , for $i \in \{red, green, blue\}$

Then according to Law of total probability:

$Pr(A) = \sum_{i \in \{red, green, blue\}}Pr(A | H_i) \cdot Pr(H_i)$

At first, we can state that $Pr(A|H_i) = Pr(H_i)$ because marble is replaced, so total count of marbles is equal when we took them first or second time.

We can calculate each of these probablilities using general formula:

$Pr(H_i) =$ (count of marbles with color i) / (total count of marbles)

$Pr(H_r) = 4 / 12 = 1/3$

$Pr(H_g) = 5/12$

$Pr(H_b) = 3/12 = 1/4$

$Pr(A) = (1/3)^2 + (5/12)^2 + (1/4)^2 = 50/144 = 25/72 = 0.35$

So, answer is 25/72 (or approximately 35%).