Question

A bag contains 3 red, 4 blue and 3 green balls. A ball is drawn at random. Event E represents `not drawing a blue ball'. Then find the probability of \(\bar{E}\) (complementary event of E).

• A. \(\dfrac{3}{5}\)
• B. \(\dfrac{3}{10}\)
• C. \(\dfrac{7}{10}\)
• D. \(\dfrac{2}{5}\)

Hint:

The key here is reading carefully: the question asks for \(P(\bar{E})\) (the complement), not \(P(E)\). Event \(E\) = ``not blue'' \(\Rightarrow\) \(\bar{E}\) = ``blue''. The complementary probability law \(P(\bar{E}) = 1 - P(E)\) is always the fastest path. Make sure you identify what \(E\) is and what \(\bar{E}\) is before computing.

Answer 1

The Correct Option is D

Step 1: Understand the complement.
Event $E$ is not drawing a blue ball. Its complement $\bar{E}$ is the opposite, which is drawing a blue ball. We need the probability of $\bar{E}$.

Step 2: Count all the balls.
There are $3$ red, $4$ blue, and $3$ green balls. The total is $3 + 4 + 3 = 10$ balls.

Step 3: Count the favourable balls.
For $\bar{E}$, drawing a blue ball, the favourable count is the number of blue balls, which is $4$.

Step 4: Write the probability.
\[ P(\bar{E}) = \frac{\text{blue balls}}{\text{total balls}} = \frac{4}{10} \]

Step 5: Simplify the fraction.
Divide top and bottom by $2$. \[ \frac{4}{10} = \frac{2}{5} \]

Step 6: State the answer.
So the probability of the complementary event is $\frac{2}{5}$. Therefore \[ \boxed{\dfrac{2}{5}} \]