A box contains 3 blue marbles, 4 red, 6 green marbles and 2 yellow marbles. If two marbles are picked at random, what is the probability that they are either blue or yellow?

Solution(By Examveda Team)

Given that there are three blue marbles, four red marbles, six green marbles and two yellow marbles.
Probability that both marbles are blue
$$\eqalign{ & = \frac{{{}^3{C_2}}}{{{}^{15}{C_2}}} \cr & = \frac{{3 \times 2}}{{15 \times 14}} \cr & = \frac{1}{{35}} \cr} $$
Probability that both are yellow
$$\eqalign{ & = \frac{{{}^2{C_2}}}{{{}^{15}{C_2}}} \cr & = \frac{{2 \times 1}}{{15 \times 14}} \cr & = \frac{1}{{105}} \cr} $$
Probability that one blue and other is yellow
$$\eqalign{ & = \frac{{{}^3{C_1} \times {}^2{C_1}}}{{{}^{15}{C_2}}} \cr & = \frac{{2 \times 3 \times 2}}{{15 \times 14}} \cr & = \frac{2}{{35}} \cr} $$
∴ Required probability
$$\eqalign{ & \frac{1}{{35}} + \frac{1}{{105}} + \frac{2}{{35}} \cr & = \frac{{3 + 1 + 6}}{{105}} \cr & = \frac{{10}}{{105}} \cr & = \frac{2}{{21}} \cr} $$