Question

From a well-shuffled pack of 52 cards one card is selected at random. Find the probability that the card is
(i) an ace of heart.
(ii) a club card.
(iii) a face card.
(iv) a queen or a king.

Answer 1

We need to find the probability for each case. We know that the number of ace, spade, hearts, and diamonds are 13 each. Each set will have 1 King, Queen, and jack. Hence, total $\text{Kings=Queens=Jacks=4}$ . Face cards include Kings, Queens, and Jacks $=4+4+4=12$. The number of Ace cards =4 (1 in each section). It is given that the total number of cards is 52. Hence, by substituting proper numbers in the formula $\text{Probability}=\dfrac{\text{Number of favorable outcome}}{\text{Total number of outcomes}}$, we will get the required solution.

Complete step-by-step solution
It is given that a card is drawn at random from a pack of 52 cards. We need to find the probability that the card is drawn is an ace of heart, a club card, a face card, a queen, or a king for each case.
Let us see the number of cards in a 52 pack of cards. We know that
Total number of cards =52
The number of Clubs card =13
Number of Spades =13
Number of Hearts =13
Number of Diamonds =13
Number of Kings =4 (1 in each section)
Number of Queens =4 (1 in each section)
Number of Jacks =4 (1 in each section)
Number of Face cards $=\text{Number of queens+kings+jacks}=4+4+4=12$
Number of Ace card =4 (1 in each section)
Now, let us find the probability for the given sections.
(i) We have to find the probability that the card is drawn is an ace of heart.
We know that probability is given by
$\begin{aligned} & \text{Probability}=\dfrac{\text{Number of favourable outcome}}{\text{Total number of outcomes}} \\\ & \Rightarrow P\left( \text{ace of heart} \right)=\dfrac{1}{52} \\\ \end{aligned}$
(ii) Let us find the probability that the card drawn is a club. That is,
$\Rightarrow P\left( \text{clubs} \right)=\dfrac{13}{52}=\dfrac{1}{4}$
(iii) We have to find the probability that the card drawn is a face card.
$\Rightarrow P\left( \text{face} \right)=\dfrac{12}{52}=\dfrac{3}{13}$
(iv) Let us find the probability that the card drawn is a king or queen.
A king or queen means we have to add the number of kings and queens. This will be the number of favourable outcomes. That is,
$\Rightarrow P\left( \text{King or Queen} \right)=\dfrac{4+4}{52}=\dfrac{8}{52}=\dfrac{2}{13}$

Note: You must know the number of cards present in the deck of 52 cards. Do not write the formula for probability as $P(A)=\dfrac{\text{Total number of outcomes}}{\text{Number of favourable outcome}}$ instead of $\text{Probability}=\dfrac{\text{Number of favourable outcome}}{\text{Total number of outcomes}}$ . Also, in the last section, we had to find the probability of the card being King or Queen. In this case, we will add the number of Kings and Queens. Do not multiply them. When it is specified as King and Queen, then only we will do the multiplication.