Q.3.48 - A First Course in Probability

Question

Each of $2$ cabinets identical in appearance has $2$ drawers. Cabinet A contains a silver coin in each drawer, and cabinet B contains a silver coin in one of its drawers and a gold coin in the other. A cabinet is randomly selected, one of its drawers is opened, and a silver coin is found. What is the probability that there is a silver coin in the other drawer? 

Step-by-Step Solution

Answer

The probability that there is a silver coin in the other drawer is $\frac{2}{3}$.

Step 1 : Given Information

A: Event that a silver coin was found in one of the drawers.

B_$1$: Event that the cabinet is A.

B_$2$: Event that the cabinet is B.

Step 2 : Explanation

Here,  $P\left(B_1\right)$$=$$P\left(B_2\right)$= $\frac{1}{2}$

          $P\left(\frac{A}{B_1}\right)=1$,$P\left(\frac{A}{B_2}\right)=\frac{1}{2}$

    $P\left(\frac{B_1}{A}\right) =\frac{P\left(B_1\right)P\left({\displaystyle \frac{A}{B_1}}\right)}{P\left(B_1\right)P\left({\displaystyle \frac{A}{B_1}}\right)+P\left(B_2\right)P\left({\displaystyle \frac{A}{B_2}}\right)}$

              $=\frac{{\displaystyle \frac{1}{2}}\times 1}{{\displaystyle \frac{1}{2}}\times 1+{\displaystyle \frac{1}{2}}\times {\displaystyle \frac{1}{2}}}$

              $$\begin{gathered} =\frac{1}{1+0.5} \\ =\frac{1}{1.5} \\ =\frac{2}{3} \end{gathered}$$

Step 3 : Final Answer

$\frac{2}{3}$