Question

Suppose we have four boxes $A,B,C$ and $D$ containing colored marbles as given :
One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box $A$, box $B$, box $C$ ?

Answer 1

Let $R$ be the event of drawing the red marble.
Let $E_A,E_B$, and $E_C$ respectively denote the events of selecting the box $A,B$, and $C$.
Total number of marbles $=40$
Number of red marbles $=15$

$\therefore P\left(R\right)=\frac{15}{40}=\frac{3}{8}$

Probability of drawing the red marble from box $A$ is given by $P\left(E_A|R\right)$.
$\therefore P\left(E_A|R\right)=\frac{P(E_A\cap R)}{P\left(R\right)}=\frac{\frac{1}{40}}{\frac{3}{8}}=\frac{1}{15}$
Probability that the red marble is from box $B$ is $P\left(E_B|R\right)$.
$\Rightarrow P\left(E_B|R\right)=\frac{P(E_B\cap R)}{P\left(R\right)}=\frac{\frac{6}{40}}{\frac{3}{8}}=\frac{2}{5}$
Probability that the red marble is from box $C$ is $P\left(E_C|R\right)$.
$\Rightarrow P\left(E_C|R\right)=\frac{P(E_C\cap R)}{P\left(R\right)}=\frac{\frac{8}{40}}{\frac{3}{8}}=\frac{8}{15}=0.53$