Question

One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting:
(a) a king of red suit
(b) a face cards
(c) a red face cards
(d) a queen of black suit
(e) a jack of hearts
(f) a spade

Answer 1

Hint:The event of getting a king or queen from a pack of 52 cards is required for solving this problem. We first have to evaluate the sample space of the particular event and then find the favorable outcome from the sample space. By taking the ratio of these events, we evaluate the probability. We follow the same methodology for solving the question.

Complete step-by-step answer:
The number of cards in pack = 52. We have chosen 1 card so selection is for 1 object.
(a) No. of kings of red suit = 2
Therefore, $^{2}{{C}_{1}}$(Selecting 1 out of 2 items) times out of ${}^{52}{{C}_{1}}$ ( Selecting 1 out of 52 items) a king of red suit is picked.
Let, ${{E}_{1}}$ be the event of getting a king of red suit from pack

We know that, Probability P(E) =$\dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$
$P\left( {{E}_{1}} \right)=\dfrac{{}^{2}{{C}_{1}}}{{}^{52}{{C}_{1}}}=\dfrac{2}{52}=\dfrac{1}{26}$
Hence, the probability of getting a king of red suits is $\dfrac{1}{26}$.


(b) Number of the face cards = 12.
Therefore, $^{12}{{C}_{1}}$(Selecting 1 out of 12 items) times out of ${}^{52}{{C}_{1}}$ ( Selecting 1 out of 52 items) a face card is picked.
Let, ${{E}_{2}}$ be the event of getting a face card from pack
By using the formula for probability, we get
\[P\left( {{E}_{2}} \right)=\dfrac{{}^{12}{{C}_{1}}}{{}^{52}{{C}_{1}}}=\dfrac{12}{52}=\dfrac{3}{13}\]
Hence, the probability of getting a face card is $\dfrac{3}{13}$.


(c) No. of red face cards = 6.
Therefore, $^{6}{{C}_{1}}$(Selecting 1 out of 6 items) times out of ${}^{52}{{C}_{1}}$ ( Selecting 1 out of 52 items) a red face card is picked.
Let, ${{E}_{3}}$ be the event of getting a red face card from the pack.

By using the formula for probability, we get
$P\left( {{E}_{3}} \right)=\dfrac{{}^{6}{{C}_{1}}}{{}^{52}{{C}_{1}}}=\dfrac{6}{52}=\dfrac{3}{26}$
Hence, the probability of getting a king of red suits is $\dfrac{3}{26}$.


(d) No. of queens of black suit =2
Therefore, $^{2}{{C}_{1}}$(Selecting 1 out of 2 items) times out of ${}^{52}{{C}_{1}}$ ( Selecting 1 out of 52 items) a queen of black suit is picked.
Let, ${{E}_{4}}$ be the event of getting a queen of black suit from pack

By using the formula for probability, we get
$P\left( {{E}_{4}} \right)=\dfrac{{}^{2}{{C}_{1}}}{{}^{52}{{C}_{1}}}=\dfrac{2}{52}=\dfrac{1}{26}$
Hence, the probability of getting a queen of black suits is $\dfrac{1}{26}$.


(e) No. of jack of heart = 1.
Therefore, $^{1}{{C}_{1}}$(Selecting 1 out of 1 item) times out of ${}^{52}{{C}_{1}}$ ( Selecting 1 out of 52 items) a jack of heart is picked.
Let, ${{E}_{5}}$ be the event of getting a jack of heart from pack
By using the formula for probability, we get
$P\left( {{E}_{5}} \right)=\dfrac{{}^{1}{{C}_{1}}}{{}^{52}{{C}_{1}}}=\dfrac{1}{52}$
Hence, the probability of getting a jack of hearts is $\dfrac{1}{52}$.


(f) No. of spades = 13.
Therefore, $^{13}{{C}_{1}}$(Selecting 1 out of 13 items) times out of ${}^{52}{{C}_{1}}$ ( Selecting 1 out of 52 items) a spade is picked.
Let, ${{E}_{6}}$ be the event of getting a spade from pack
By using the formula for probability, we get
$P\left( {{E}_{6}} \right)=\dfrac{{}^{13}{{C}_{1}}}{{}^{52}{{C}_{1}}}=\dfrac{13}{52}=\dfrac{1}{4}$
Hence, the probability of getting a spade is $\dfrac{1}{4}$.

Note: The key concept for solving a problem is the knowledge of probability of occurrence of an event. Students must be careful while calculating the space for favourable events. There should be no redundancy of particular events in the favourable outcomes.Student should know this concept while solving these types of problems.A standard deck of playing cards contains 52 cards.Divided equally into two colors "Red" and "Black".Deck of 52 cards has four suits "Spades", "Hearts", "Diamonds" and "Clubs".Hearts and Diamonds comes in Red color and Spades and Clubs comes in Black color.Each 4 suits contains 13 cards : Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King.A king, queen, or jack of a deck of playing cards are known as face cards.