If two cards are drawn randomly from a pack of 52 playing cards, then the mean of the probability…

If two cards are drawn randomly from a pack of 52 playing cards, then the mean of the probability distribution of number of kings is
  1. $\frac{215^{\prime}}{221}$
  2. $\frac{2}{13}$
  3. $\frac{188}{221}$
  4. $\frac{13}{2}$

Solution

Let $X$ : number of king card So, $P(X=0)=\frac{{ }^{48} C_2}{{ }^{52} C_2}, P(X=1)=\frac{{ }^4 C_1{ }^{48} C_1}{{ }^{52} C_2}$ and $P(X=2)=\frac{{ }^4 C_2}{{ }^{52} C_2}$ Now, mean of probability distribution $\begin{aligned} & =0 \times \frac{{ }^{48} C_2}{{ }^{52} C_2}+1 \times \frac{{ }^4 C_1 \times{ }^{48} C_1}{{ }^{52} C_2}+\frac{2 \times{ }^4 C_2}{{ }^{52} C_2} \\ & =0+\frac{4 \times 48 \times 2}{52 \times 51}+\frac{2 \times 4 \times 3 \times 2}{2 \times 52 \times 51} \\ & =\frac{32}{221}+\frac{2}{221}=\frac{34}{221}=\frac{2}{13} \end{aligned}$

Asked in: AP EAMCET 2024 (19 May Shift 2)

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