Question

A box contains $10$ red marbles, $20$ blue marbles and $30$ green marbles. $5$ marbles are drawn at random. From the box, what is the probability that:
(i) all are blue?
(ii) at least one is green?

Answer 1

Out of $60$ marbles, five marbles can be drawn in ${}^{60}{C}_5$ ways
∴ Total number of elementary events $={}^{60}{C}_5$
(i)
Out of $2$0 blue marbles, five blue marbles can be chosen in ${}^{20}{C}_5$ ways
∴ Favourable number of events $={}^{20}{C}_5$. ways

Probability of an event $=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$
Hence, the required probability is given by

$=\frac{{}^{20}{C}_5}{{}^{60}{C}_5}$

(ii)

Here, No green $=10+20=30$

favourable outcomes $={}^{30}{C}_5$

Total number of outcomes ${}^{60}{C}_5$

Probability of no green $=\frac{{}^{30}{C}_5}{{}^{60}{C}_5}$
Thus, P(at least one green) $=1–P\left(\text{no green}\right)$

$=1-{}^{60}{C}_5$