Question

There are two balls in an urn whose colors are not known ( ball can be either white or black). A white ball is put into the urn. A ball is then drawn from the urn. The probability that it is white is

• A. $\frac{1}{4}$
• B. $\frac{1}{3}$
  
• C. $\frac{2}{3}$
  
• D. $\frac{1}{6}$

Answer 1

Answer: (C)

$\frac{2}{3}$

Let $E_i(0\le i\le 2)$ denotes the event that urn contains $i$ white and $2-i$ black balls.

Let $A$ denotes the event that a white ball is drawn from the urn.

We have $P\left(E_i\right)=\frac{1}{3}$ for $i=0,1,2$ and $P\left(\frac{A}{E_i}\right)=\frac{1}{3},P\left(\frac{A}{E_2}\right)=\frac{2}{3},P\left(\frac{A}{E_3}\right)=1$

By the total probability rule,

$P\left(A\right)=P\left(E_1\right)P\left(\frac{A}{E_1}\right)+P\left(E_2\right)P\left(\frac{A}{E_2}\right)+P\left(E_3\right)P\left(\frac{A}{E_3}\right)$

$=\frac{1}{3}[\frac{1}{3}+\frac{2}{3}+1]=\frac{2}{3}$