Question

A card is drawn at random from a well shuffled pack of $52$ cards.

The probability of getting a two of heart or two of diamonds is

• A. $\frac{1}{26}$
• B. $\frac{2}{17}$
• C. $\frac{1}{13}$
• D. None of these

Answer 1

The correct option is B

$\frac{2}{17}$

Explanation for the correct option:

Step 1. Find the probability of getting a $2$ of heart or $2$ of diamonds:

Total number of cards $=52$

total number of favorable outcomes $=2$ of heart or $2$ of diamond

$$\begin{gathered} =1+1 \\ =2 \end{gathered}$$

$(\because$in a deck of cards there can only be one $2$ of heart and only one $2$ of diamond$)$

we have to choose only one card

So, favorable outcomes, $n\left(E\right)={}^{13}C_2+{}^{13}C_2$

and Total possible outcomes, $n\left(S\right)={}^{52}C_2$

Step 2. Finding the required probability

$P\left(E\right)=\frac{{}^{13}C_2+{}^{13}C_2}{{}^{52}C_2}$ $\left[P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}\right]$

$=\frac{78+78}{1326}$ $\left[\mathbf{\because }{}^{\mathbf{n}}\mathit{C}_{\mathbf{r}}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{n}\mathbf{!}}{\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{!}\mathbf{r}\mathbf{!}}\right]$

$=\frac{156}{1326}$

$=\mathbf{}\frac{\mathbf{2}}{\mathbf{17}}$

Hence, Option ‘B’ is Correct.