Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Let X denote the…
Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Let X denote the random variable of number of jacks obtained in the two drawn cards. Then $\mathrm{P}(\mathrm{X}=1)+\mathrm{P}(\mathrm{X}=2)$ equals
$\frac{24}{169}$
$\frac{52}{169}$
$\frac{25}{169}$
$\frac{49}{169}$
Solution
Since two cards are drawn successively with replacement, we get
$P(X=1)=2 \times \frac{{ }^4 C_1 \times{ }^{48} C_1}{{ }^{52} C_1 \times{ }^{52} C_1}=2 \times \frac{4 \times 48}{52 \times 52}=\frac{24}{169}$
$\begin{aligned} & P(X=2)=\frac{{ }^4 C_1 \times{ }^4 C_1}{{ }^{52} C_1 \times{ }^{51} C_1}=\frac{4 \times 4}{52 \times 52}=\frac{1}{169} \\ \therefore \quad & P(X=1)+P(X=2)=\frac{25}{169}\end{aligned}$