Question

From a pack of 52 playing cards, two cards are drawn together at random. Calculate the probability of both the cards being the Kings.
A. $\dfrac{1}{{15}}$
B. $\dfrac{{25}}{{57}}$
C. $\dfrac{{35}}{{256}}$
D. None of the above

Answer 1

Hint: Here we need to find the probability that the two drawn cards are kings. So we will first find the number of ways to draw two cards from the pack of 52 cards using the permutation and combination method and then we will similarly find the number of ways to draw two of the king. Then at last we will find the required probability by taking the ratio of both of them.

Complete step by step solution:
Here we need to find the probability that the two drawn cards are kings. We know that there are a total of 52 cards and there are a total of 4 cards of kings in a pack of cards.
Now, we will find the number of ways to draw two cards from the pack of 52.
Two cards can be drawn from a pack of 52 cards in ${}^{52}{C_2}$ ways.
We know the formula of combination that
${}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)! \times r!}}$
Using this formula of combination, we get
Number of ways to draw two cards from the pack of 52 cards $ = \dfrac{{52!}}{{50! \times \left( {52 - 2} \right)!}}$
On putting the value of factorials, we get
A number of ways to draw two cards from the pack of 52 cards $ = \dfrac{{52 \times 51 \times 50!}}{{2! \times 50!}} = 26 \times 51$
On multiplying the numbers, we get
Number of ways to draw two cards from the pack of 52 cards $ = 1326$
Now, we will find the number of ways to draw two cards of kings from the pack of 52.
Two cards of cards can be drawn in ${}^4{C_2}$ ways.
We know the formula of combination that
${}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)! \times r!}}$
Using this formula of combination, we get
A number of ways to draw two cards of kings $ = \dfrac{{4!}}{{2! \times \left( {4 - 2} \right)!}}$
On putting the value of factorials, we get
Number of ways to draw two cards of kings $ = \dfrac{{4 \times 3 \times 2!}}{{2 \times 1 \times 2!}} = 2 \times 3$
On multiplying the numbers, we get
Number of ways to draw two cards of kings $ = 6$
Now, we will find the probability that the two drawn cards are kings and this will be equal to the ratio of the number of ways to draw two cards of kings to the number of ways to draw two cards from the total cards.
Required probability $ = \dfrac{6}{{1326}}$
We can see that the obtained dfraction can be reduced further.
Required probability $ = \dfrac{1}{{221}}$

Therefore, the correct option is option D.

Note: To solve such a type of problem, we need to know about probability and its properties. Probability is defined as the ratio of the number of desired outcomes to the total number of possible outcomes. Remember that the value of probability cannot be greater than 1 and the value of probability cannot be negative. Also, remember that the probability of a sure event is always one.