Question

The bag ${}^{\prime }{A}^{\prime }$ contains $5$ white and $3$ black balls while the bag ${}^{\prime }{B}^{\prime }$ contains $4$ white and $7$ black balls. One of the bags is chosen at random and a ball is drawn from it. What is the probability that the ball is white?

• A. $\frac{87}{176}$
• B. $\frac{81}{172}$
• C. $\frac{82}{179}$
• D. $\frac{87}{179}$

Answer 1

Answer: (A)

Let $E_1$ be the event that bag $A$ is chosen,
$E_2$ be the event that bag $B$ is chosen and
$A$ be the event that white ball is drawn
Note that $E_1$ and $E_2$ are mutually exclusive and exhaustive events.
Since one of the bag is chosen at random,
$P\left(E_1\right)=\frac{1}{2}$, $P\left(E_2\right)=\frac{1}{2}$
$P\left(A|E_1\right)=$ probability of drawing a white ball from bag $A=\frac{5}{8}$
$P\left(A|E_2\right)=$ probability of drawing a white ball from bag $B=\frac{4}{11}$
By using law of total probability, we get
$P\left(A\right)=P\left(E_1\right)P\left(A|E_1\right)+P\left(E_2\right)P\left(A|E_2\right)$
$=\frac{1}{2}\cdot \frac{5}{8}+\frac{1}{2}\cdot \frac{4}{11}$
$=\frac{1}{2}\left(\frac{5}{8}+\frac{4}{11}\right)$
$=\frac{1}{2}\cdot \frac{87}{88}=\frac{87}{176}$