Question

There is a bag filled with 3 blue and 5 red marbles.

A marble is taken at random from the bag, the colour is noted and then it is not replaced.
Another marble is taken at random.
What is the probability of getting 2 of the same colour?

Answer 1

53.57%

Let us label the three blue marbles b1, b2 and b3, and the five red marbles r1, r2, r3, r4 and r5.

Let marble A be the first marble taken from the bag and let marble B be the second marble taken from the bag.

In this answer, I will use the following terminology:

The identity of a mable refers to which of the 8 individual marbles it is. Thus b1 and b2 are two possible identities for marble A.

An outcome refers to the particular pairs of identities for marbles A and B. This, for instance, marble A being r1 and marble B being b2 is one possible outcome.

The total number of marbles is 3+5 = 8. So marble A could be any one of those 8 possible marbles. After marble A is taken from the bag, there are 7 marbles remaining in the bag, any one of which could be marble B. So for each of the 8 possible identities of marble A, there are 7 possible identities for marble B. So the number of outcomes is 8×7 = 56.

There are two possible sets of outcomes which correspond with the two marbles having different colours; those outcomes in which marble A is blue and marble B is red, and those outcomes in which marble A is red and marble B is blue.

The number of possible identities for marble A which is a blue marble is 3. If marble A is blue, then for each of those 3 possible identities for marble A which is a blue marble, there are 5 possible identities for marble B which is a red marble. So there are 3×5 = 15 possible outcomes in which marble A is blue and marble A is red.

The number of possible identities for marble A which is a red marble is 5. If marble A is red, then for each of those 5 possible identities for marble A which is a red marble, there are 3 possible identities for marble B which is a blue marble. So there are 5×3 = 15 possible outcomes in which marble A is red and mable A is blue.

Therefore the total number of possible outcomes which result in the two marbles taken from the bag being different colours is 15+15 = 30.

Therefore the probability that the two marbles are different colours is equal to (the number of possible outcomes which are consistent with the two marbles being of different colours) divided by (the number of all possible outcomes), which is 30/56 = 15/28 ≅ 0.5357 = 53.57%..