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. $\frac{3}{4}$

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

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

  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

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

$=\frac{39}{50}$

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