Question

What is the probability of drawing a king and a queen consecutively from a deck of 52 cards, without replacement? Choose the correct option.
A. \[\dfrac{4}{51}\]
B. \[\dfrac{1}{13}\]
C. \[\dfrac{2}{663}\]
D. \[\dfrac{4}{663}\]

Answer 1

Hint: In a deck of 52 cards there are 4 kings and 4 queens. The probability of getting king first and then queen consecutively without replacement is to be found next, by applying the probability formula \[\text{Probability=}\dfrac{\text{No}\text{.}\,\text{of}\,\text{Favourable}\,\text{cases}}{\text{Total}\,\text{cases}}\].

Complete step-by-step answer:
replacement from a deck of 52 cards.
So here we know that there are 52 cards out of which 4 are king and 4 are queens. So in the first draw the probability of getting the king will be found using the probability formula\[\text{Probability=}\dfrac{\text{No}\text{.}\,\text{of}\,\text{Favourable}\,\text{cases}}{\text{Total}\,\text{cases}}\], where the number of favourable cases is 4 and total case is 52.
So the probability of getting the king in the first draw is \[\dfrac{\text{4}}{\text{52}}\].
Next, in the second draw we want the queen to come and also there is no replacement done, so the number of cards that are left will be 51 now, and there are 4 queens in this 51 cards. So the probability of getting a queen in the second draw is \[\dfrac{\text{4}}{\text{51}}\].
Now the probability of drawing a king and a queen consecutively will be the product of these two probabilities. So the required probability is found as follows:
\[\begin{align}
  & \Rightarrow \dfrac{\text{4}}{\text{52}}\times \dfrac{\text{4}}{\text{51}} \\
 & \Rightarrow \dfrac{4}{663} \\
\end{align}\]
Hence the correct answer is option D) \[\dfrac{4}{663}\]

Note: Here we are drawing the king and then the queen so this is a consecutive event. So in the first draw there are 52 cards and in the consecutively second draw there will be just 51 cards left. So we have to keep that in mind while finding the probability.