Question

A box contains $5$ silver coins and $4$ gold coins, two coins are stolen from the box. Now one coin is drawn at random from the box and is found to be of gold. Find the probability that the lost coins also were of gold.

Answer 1

Box $I:2$ Silver coins $+4$ gold coins

Box $II:4$ Silver coins $+3$ gold coins

Let $E_1$ be the event of selecting a gold coin from box $I$

and $E_2$ be the event of selecting a gold coin from box $II.$

$A:$ Probability of gold coin

$\Rightarrow P\left(E_1\right)={}^6{C}_4$ $P\left(E_2\right)={}^7{C}_3$

$\Rightarrow P\left(A/E_1\right)=P\left(A/E_2\right)=\frac{1}{2}$

$\Rightarrow P\left(E/E_1\right)=\frac{P\left(E_1\right)P\left(A/E_1\right)}{P\left(E_1\right)P\left(A/E_1\right)P\left(E_2\right)P\left(A/E_2\right)}$

$=\frac{{}^6{C}_4\times 1/2}{{}^6{C}_4\times 1/2+{}^7{C}_3\times 1/2}$

$=\frac{7.5}{7.5+17.5}$

$=0.34$

Hence, the answer is $0.34$