# A pack of cards has one card missing.

Question:

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :

1. (1) $\frac{3}{4}$

2. (2) $\frac{52}{867}$

3. (3) $\frac{39}{50}$

4. (4) $\frac{22}{425}$

Correct Option: , 3

Solution:

$\mathrm{E}_{1}$ : Event denotes spade is missing

$\mathrm{P}\left(\mathrm{E}_{1}\right)=\frac{1}{4} ; \mathrm{P}\left(\overline{\mathrm{E}}_{1}\right)=\frac{3}{4}$

A: Event drawn two cards are spade

$\mathrm{P}(\mathrm{A})=\frac{\frac{1}{4} \times\left(\frac{12 \mathrm{C}_{2}}{51 \mathrm{C}_{2}}\right)+\frac{3}{4} \times\left(\frac{13 \mathrm{C}_{2}}{51 \mathrm{C}_{2}}\right)+\frac{3}{4} \times\left(\frac{13 \mathrm{C}_{2}}{51 \mathrm{C}_{2}}\right)}{\frac{1}{4} \times\left(\frac{12 \mathrm{C}_{2}}{51 \mathrm{C}_{2}}\right)+\frac{3}{4} \times\left(\frac{13 \mathrm{C}_{2}}{51 \mathrm{C}_{2}}\right)}$

$=\frac{39}{50}$