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?

Answer 1

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 $=\frac{{}^{20}{C}_5}{{}^{60}{C}_5}$
(ii) Number of ways in which the drawn marble is not green $={}^{(20+10)}{C}_5{=}^{30}{C}_5$
$\Rightarrow$ Probability that no marble is green $=\frac{{}^{30}{C}_5}{{}^{60}{C}_5}$
$\therefore$ Probability that at least one marble is green $=1-\frac{{}^{30}{C}_5}{{}^{60}{C}_5}$