Probability 4 Question 2
2. A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is
(2018 Main)
(a) $\frac{3}{10}$
(b) $\frac{2}{5}$
(c) $\frac{1}{5}$
(d) $\frac{3}{4}$
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Answer:
Correct Answer: 2. (b)
Solution:
- Key idea Use the theorem of total probability
Let $E _1=$ Event that first ball drawn is red $E _2=$ Event that first ball drawn is black $A=$ Event that second ball drawn is red
$$ \begin{aligned} & P\left(E _1\right)=\frac{4}{10}, P \frac{A}{E _1}=\frac{6}{12} \\ \Rightarrow \quad & P\left(E _2\right)=\frac{6}{10}, P \frac{A}{E _2}=\frac{4}{12} \end{aligned} $$
By law of total probability
$$ \begin{aligned} P(A) & =P\left(E _1\right) \times P \frac{A}{E _1}+P\left(E _2\right) \times P \frac{A}{E _2} \\ & =\frac{4}{10} \times \frac{6}{12}+\frac{6}{10} \times \frac{4}{12}=\frac{24+24}{120}=\frac{48}{120}=\frac{2}{5} \end{aligned} $$
