From a pack of $52$ cards, two cards are drawn together at random. What is the probability of both the cards being kings ?

### Concept & Combinations in Probability When selecting multiple items simultaneously, we use combinations to find the total number of ways to choose. The probability is the number of favorable combinations divided by total combinations. $$\text{Probability} = \frac{\binom{\text{Favorable}}{\text{Chosen}}}{\binom{\text{Total}}{\text{Chosen}}}$$ ### Step-by-Step Solution * **Total Outcomes:** Number of ways to draw $2$ cards from $52$. * $n(S) = \binom{52}{2} = \frac{52 \times 51}{2 \times 1} = 26 \times 51 = 1326$. * **Favorable Outcomes:** Number of ways to draw $2$ kings from the $4$ available kings. * $n(E) = \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 2 \times 3 = 6$. * **Calculation:** $P(E) = \frac{n(E)}{n(S)} = \frac{6}{1326}$. * Divide both numerator and denominator by $6$: $\frac{1}{221}$. ### Exam Strategy & Shortcut Instead of combinations, use dependent probabilities. The probability of the first card being a king is $\frac{4}{52}$. The probability of the second card being a king (given the first was a king) is $\frac{3}{51}$. Multiply them: $\frac{4}{52} \times \frac{3}{51} = \frac{1}{13} \times \frac{1}{17} = \frac{1}{221}$. This is much faster. ### Common Pitfall Assuming the draws are with replacement, which would lead to calculating $(\frac{4}{52}) \times (\frac{4}{52})$, an incorrect approach when cards are drawn together. ### Final Answer Therefore, the correct answer is **$\frac{1}{221}$**.