Question

A pack of cards was found to contain only $51cards$. If first $13cards$, which are examined, are all red, then find the probability that the missing cards is black?

• A. $2/3$
• B. $1/3$
• C. $2/9$
• D. $1/6$

Answer 1

Answer: (A)

$2/3$

Let $A=$ Missing card is black.
$B=$ Missing card is red.
$D=$ First $13$ cards found are red.
$P\left(\frac{A}{D}\right)=\frac{P\left(\frac{D}{A}\right)P\left(A\right)}{P\left(\frac{D}{A}\right)P\left(A\right)+P\left(\frac{D}{B}\right)P\left(B\right)}$

$=\frac{\frac{26}{51}\times \frac{25}{50}\times ...\times \frac{14}{39}\times \frac{1}{2}}{\frac{26}{51}\times \frac{25}{50}\times ...\times \frac{14}{39}\times \frac{1}{2}+\frac{25}{51}\times \frac{24}{50}\times ...\times \frac{13}{39}\times \frac{1}{2}}$

$=\frac{26}{26+13}=\frac{2}{3}$