# Two cards are drawn successively with replacement

Question:

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let $X$ denote the random variable of number of aces obtained in the two drawn cards. Then $\mathrm{P}(\mathrm{X}=1)+\mathrm{P}(\mathrm{X}=2)$ equals :

1. $52 / 169$

2. $25 / 169$

3. $49 / 169$

4. $24 / 169$

Correct Option: , 2

Solution:

Two cards are drawn successively with replacement

4 Aces 48 Non Aces

$\mathrm{P}(\mathrm{x}=1)=\frac{{ }^{4} \mathrm{C}_{1}}{{ }^{52} \mathrm{C}_{1}} \times \frac{48 \mathrm{C}_{1}}{52 \mathrm{C}_{1}}+\frac{48 \mathrm{C}_{1}}{52 \mathrm{C}_{1}} \times \frac{4 \mathrm{C}_{1}}{52 \mathrm{C}_{1}}=\frac{24}{169}$

$\mathrm{P}(\mathrm{x}=2)=\frac{{ }^{4} \mathrm{C}_{1}}{{ }^{52} \mathrm{C}_{1}} \times \frac{{ }^{4} \mathrm{C}_{1}}{{ }^{52} \mathrm{C}_{1}}=\frac{1}{169}$

$P(x=1)+P(x=2)=\frac{25}{169}$