A basket contains 4 red balls and 5 blue balls. One ball drawn and laid aside, and a second ball is drawn. Draw a tree diagram to represent the possible outcomes and probabilities. Use the tree diagram to answer the following questions.
(i) What is the probability of drawing a red ball and then a blue ball?
(ii) What is the probability of drawing 2 blue balls?

SOLUTION

Understand the experiment

The basket contains:
(i) 4 red balls(R)
(ii)5 blue balls(B)
Total balls: \(4 + 5 = 9\)
One ball is drawn and kept aside. Then a second ball is drawn. So this experiment is without replacement.

Draw the tree diagram

Write the outcome probabilities

(1) Red then Red(RR):
\(P(RR) = \frac{4}{9} × \frac{3}{8} = \frac{12}{72} = \frac{1}{6}\)
(2) Red then Blue(RB):
\(P(RB) = \frac{4}{9} × \frac{5}{8} = \frac{20}{72} = \frac{5}{18}\)
(3) Blue then Red(BR):
\(P(BR) = \frac{5}{9} × \frac{4}{8} = \frac{20}{72} = \frac{5}{18}\)
(4) Blue then Blue(BB):
\(P(BB) = \frac{5}{9} × \frac{4}{8} = \frac{20}{72} = \frac{5}{18}\)

(i) Probability of drawing a Red ball and then a Blue ball

\(P(RB) = \frac{4}{9} × \frac{5}{8} = \frac{5}{18}\)

(ii) Probability of drawing two Blue balls

\(P(BB) = \frac{5}{9} × \frac{4}{8} = \frac{5}{18}\)

🏆

Final Answer : (i) P(RB) = \(\frac{1}{18}\)
(ii) P(BB) = \(\frac{5}{18}\)

Concept Note

A tree diagram helps show all possible outcomes step by step.
Since the first ball is not replaced, the total number of balls decreases after the first draw.
For successive events, probabilities are multiplied.