Question

Two cards are drawn together from a pack of 52 cards. The probability that one is a spade and one is a heart is
(A) $\dfrac{3}{{20}}$
(B) $\dfrac{{29}}{{34}}$
(C) $\dfrac{{47}}{{100}}$
(D) $\dfrac{{13}}{{102}}$

Answer 1

Hint:To find the solution of the given problem, first find the total number of sample space and then find the number of ways of choosing 1 spade out of 13 and 1 heart out of 13 from the deck of 52 cards.

Complete step-by-step answer:
We know that in the deck of 52 cards there are 13 spades, and 13 heart cards. Two cards are chosen out of 52 cards. So first find the number of sample space.
Let us assume S be the sample space and then calculate the total number of sample space by using a combination method.
Now, find the number of sample space by choosing 2 cards drawn from pack of 52 cards,
\[
  n\left( S \right){ = ^{52}}{C_2} \\
   = \dfrac{{52!}}{{\left( {52 - 2} \right)!\left( {2!} \right)}} \\
   = \dfrac{{52 \times 51}}{{2 \times 1}} \\
   = 1326\]
After calculating the total sample space, find the number of ways of choosing $1$ spade out of $13$ and $1$ heart out of $13$.
\[
  n\left( E \right){ = ^{13}}{C_1}{ \times ^{13}}{C_1} \\
   = \dfrac{{13!}}{{\left( {13 - 1} \right)!\left( {1!} \right)}} \times \dfrac{{13!}}{{\left( {13 - 1} \right)!\left( {1!} \right)}} \\
   = 13 \times 13 \\
   = 169\]
The formula of the probability of an event is written as,
$P\left( A \right) = \dfrac{{n\left( E \right)}}{{n\left( S \right)}}$
Here, the number of sample space are $1326$ and the number of ways of choosing $1$ spade out of $13$ and $1$ heart out of $13$ are $169$.
So, the probability that one is a spade and one is a heart is calculated as,
$
  P\left( A \right) = \dfrac{{{\text{169}}}}{{{\text{1326}}}} \\
   = \dfrac{{13}}{{102}}$
Therefore, the probability that one is a spade and one is a heart is $\dfrac{{13}}{{102}}$.

So, the correct answer is “Option D”.

Note:While solving the sample space, make sure that the formula used for the combination is correct and be careful about the total number of spades and hearts in a deck of cards.A standard deck of playing cards consists of 52 cards. All cards are divided into 4 suits. There are two black suits — spades and clubs , two red suits — hearts and diamonds. In each suit there are 13 cards including a 2, 3, 4, 5, 6, 7, 8, 9, 10, a jack, a queen, a king and an ace.