Let us say that you have a box containing 3 red pens, 4 black pens and 2 green pens. You pick a pen (without looking) from the box and put it back. Then your friend does the same.

(A) What are the possible outcomes of the pen colours? Can you draw a tree diagram representing the possible outcomes?

(B) Can you use the tree diagram to guess the probability that both you and your friend pick pens of the same colour?                                                                                                                                                                                                        [Page No. 169]

(A) Total pens = 9.

Each colour's probability on a single draw:

$P(\text{Red}) = \frac{3}{9} = \frac{1}{3}$; $P(\text{Black}) = \frac{4}{9}$; $P(\text{Green}) = \frac{2}{9}$.

Tree diagram has $3 \times 3 = 9$ pairs.

(B) Now,

$$P(\text{Red}, \text{Red}) = \frac{3}{9} \times \frac{3}{9} = \frac{9}{81}$$

$$P(\text{Black}, \text{Black}) = \frac{4}{9} \times \frac{4}{9} = \frac{16}{81}$$

$$P(\text{Green}, \text{Green}) = \frac{2}{9} \times \frac{2}{9} = \frac{4}{81}$$

Thus, Probability of getting same colour =

$P(\text{Red}, \text{Red}) + P(\text{Black}, \text{Black}) + P(\text{Green}, \text{Green})$

$$= \frac{9}{81} + \frac{16}{81} + \frac{4}{81}$$

$$= \frac{29}{81}$$