Question

From a pack of $52$cards, two cards are drawn one by one without replacement. The probability that first drawn card is a king and second is a queen, is
$A)\dfrac{2}{{13}}$
$B)\dfrac{8}{{663}}$
$C)\dfrac{4}{{663}}$
$D)\dfrac{{103}}{{633}}$

Answer 1

First, we need to know the concept of probability.
Probability is the term mathematically with events that occur, which is the number of favorable events that divides the total number of outcomes.
In a pack of $52$cards, there are a total $4$set of different shapes like Clubs, Diamond, Hearts, and Spades.
In each of the shapes there are total $13$cards with different numbers like $1,2,3,....,10,J,Q,K$
Formula used:
$P = \dfrac{F}{T}$where P is the probability, F is the possible favorable events and T is the total outcomes from the given.

Complete step by step answer:
Since from the given question, the total card is $52$and in that the different shapes are $4$set.
In each of the sets, the cards king and queen are contained like in Heart cards we have different numbers like $1,2,3,....,10,J,Q,K$. So, in the Heart cards, we have one king and one queen.
Similarly in the cards Clubs, Diamonds and Spades we also have each King and Queen.
Thus, in totally different shapes are $4$set, we have $4$kings and $4$Queens.
Hence the favorable events for the King and Queen are $4$
Now by the probability formula, $P = \dfrac{F}{T}$where F is the possible favorable events and T is the total outcomes from the given. Thus, the total event is $52$(total cards)
Hence, the probability of King is in the first card is $P(K) = \dfrac{F}{T} \Rightarrow \dfrac{4}{{52}} = \dfrac{1}{{13}}$
But, for the Queen, we have a condition that the first card is king and then only the second card is queen.
Hence the total event for getting the second card Queen (except the first card without replacement) is $51$(total cards for Queen)
Therefore, the probability of second card Queen is $P(Q) = \dfrac{F}{T} \Rightarrow \dfrac{4}{{51}}$
Finally, combine the two values as the probability of getting the first card as King and then the second card as Queen without replacement is $P(K) \times P(Q) = \dfrac{1}{{13}} \times \dfrac{4}{{51}}$
Thus, solving this we get $P(K) \times P(Q) = \dfrac{1}{{13}} \times \dfrac{4}{{51}} \Rightarrow \dfrac{4}{{663}}$(by multiplication operation)

So, the correct answer is “Option C”.

Note: Since from the given question they said without replacement of the cards, so only if the first card is King then there is no possibility that the card is repeated, and hence we get the $52 - 1$possible ways for the Queen as the second card.
If there is no restriction like the first card need to king and the second card need to queen without replacement then we get $P(K) \times P(Q) = \dfrac{4}{{52}} \times \dfrac{4}{{52}}$(both total and favorable events are the same if there is no restriction)