What is the probability that both red and blue marbles were removed from the bag?

Question

There are 3 red and 4 blue marbles in a bag. Simultaneously at random, 3 marbles are removed from the bag. If at least 2 of the marbles removed from the bag are blue, then what is the probability that both red and blue marbles were removed from the bag?

Answer 1

We know 2 or 3 of the marbles removed were blue.

Number of ways , all blue - 1

Number of ways, 2 blue, 1 red. The blues can be chosen in 4 C 2 ways and the red in 3 C 1 ways.

So number of ways = 4 C 2 x 3 C 1 = 6 x 3 = 18

So there are 1 + 18 = 19 ways we can have at least 2 of the marbles being blue

Of these ways 18 have a red marble

So 18/19 is required probability