A box contains 10 red marbles, 20 blue marbles and 30 green marbles.

Question:

A box contains 10 red marbles, 20 blue marbles and 30 green marbles. 5 marbles are drawn from the box, what is the probability that

(i) all will be blue?

(ii) atleast one will be green?

Solution:

Total number of marbles = 10 + 20 + 30 = 60

Number of ways of drawing 5 marbles from 60 marbles $={ }^{60} C_{5}$

(i) All the drawn marbles will be blue if we draw 5 marbles out of 20 blue marbles.

5 blue marbles can be drawn from 20 blue marbles in ${ }^{20} C_{5}$ ways.

$\therefore$ Probability that all marbles will be blue $=\frac{{ }^{20} \mathrm{C}_{5}}{{ }^{60} \mathrm{C}_{5}}$

(ii) Number of ways in which the drawn marble is not green $={ }^{(20+10)} \mathrm{C}_{5}={ }^{30} \mathrm{C}_{5}$

$\therefore$ Probability that no marble is green $=\frac{{ }^{30} \mathrm{C}_{5}}{{ }^{60} \mathrm{C}_{5}}$

:Probability that at least one marble is green $=1-\frac{{ }^{30} \mathrm{C}_{5}}{{ }^{50} \mathrm{C}_{5}}$