# One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting

$\left( {\text{i}} \right)$ a king of red colour $\left( {{\text{ii}}} \right)$ a face card $\left( {{\text{iii}}} \right)$ a red face card $\left( {{\text{iv}}} \right)$ the jack of hearts $\left( {\text{v}} \right)$ a spade $\left( {{\text{vi}}} \right)$ the queen of diamonds.

Answer

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Hint- Here, we will be using the general formula for finding the probability of occurrence of an event.

Given, one card is drawn from a well-shuffled deck of 52 cards

Total number of cards$ = 52$

As we know that the general formula for probability is given by

Probability of occurrence of an event $ = \dfrac{{{\text{Number of favorable outcomes}}}}{{{\text{Total number of possible outcomes}}}}$

$\left( {\text{i}} \right)$ In this case, the favorable event is drawing a king of red colour from the deck of 52 cards.

Since we know that in a deck of 52 cards, there are 2 kings of red colour (one king of diamond and the other king of heart).

Number of kings of red colour$ = 2$

Therefore, probability of getting a king of red colour $ = \dfrac{{{\text{Number of kings of red colour}}}}{{{\text{Total number of cards}}}} = \dfrac{2}{{52}} = \dfrac{1}{{26}}$

$\left( {{\text{ii}}} \right)$ In this case, the favorable event is drawing a face card (king, queen or jack) from the deck of 52 cards.

Since we know that in a deck of 52 cards, there are a total 12 face cards (3 face cards each of heart, diamond, spade and club).

Number of face cards$ = 12$

Therefore, probability of getting a face card$ = \dfrac{{{\text{Number of face cards}}}}{{{\text{Total number of cards}}}} = \dfrac{{12}}{{52}} = \dfrac{3}{{13}}$.

$\left( {{\text{iii}}} \right)$ In this case, the favorable event is drawing a red face card (king, queen or jack) from the deck of 52 cards.

Since we know that in a deck of 52 cards, there are a total 6 red face cards (3 face cards of heart and 3 face cards of diamond).

Number of face cards$ = 6$

Therefore, probability of getting a red face card$ = \dfrac{{{\text{Number of red face cards}}}}{{{\text{Total number of cards}}}} = \dfrac{6}{{52}} = \dfrac{3}{{26}}$.

$\left( {{\text{iv}}} \right)$ In this case, the favorable event is drawing a jack of hearts card from the deck of 52 cards.

Since we know that in a deck of 52 cards, there is only 1 jack of hearts card.

Number of jack of hearts card$ = 1$

Therefore, probability of getting a jack of hearts card$ = \dfrac{{{\text{Number of jack of hearts card}}}}{{{\text{Total number of cards}}}} = \dfrac{1}{{52}}$.

$\left( {\text{v}} \right)$ In this case, the favorable event is drawing a spade card from the deck of 52 cards.

Since we know that in a deck of 52 cards, there are 13 spade cards.

Number of spade cards$ = 13$

Therefore, probability of getting a spade card$ = \dfrac{{{\text{Number of spade cards}}}}{{{\text{Total number of cards}}}} = \dfrac{{13}}{{52}} = \dfrac{1}{4}$.

$\left( {{\text{vi}}} \right)$ In this case, the favorable event is drawing a queen of diamonds card from the deck of 52 cards.

Since we know that in a deck of 52 cards, there is only 1 queen of diamonds card.

Number of queen of diamonds card$ = 1$

Therefore, probability of getting a queen of diamonds card$ = \dfrac{{{\text{Number of queen of diamonds card}}}}{{{\text{Total number of cards}}}} = \dfrac{1}{{52}}$.

Note- In these types of problems, we should know that in a deck of 52 cards there are 13 cards each of heart, diamond, spade and club. 13 cards of heart and 13 cards of diamond are red in colour whereas 13 cards of spade and 13 cards of club are black in colour. In these pairs of 13 cards there are 3 face cards consisting of a king, a queen and a jack.

Given, one card is drawn from a well-shuffled deck of 52 cards

Total number of cards$ = 52$

As we know that the general formula for probability is given by

Probability of occurrence of an event $ = \dfrac{{{\text{Number of favorable outcomes}}}}{{{\text{Total number of possible outcomes}}}}$

$\left( {\text{i}} \right)$ In this case, the favorable event is drawing a king of red colour from the deck of 52 cards.

Since we know that in a deck of 52 cards, there are 2 kings of red colour (one king of diamond and the other king of heart).

Number of kings of red colour$ = 2$

Therefore, probability of getting a king of red colour $ = \dfrac{{{\text{Number of kings of red colour}}}}{{{\text{Total number of cards}}}} = \dfrac{2}{{52}} = \dfrac{1}{{26}}$

$\left( {{\text{ii}}} \right)$ In this case, the favorable event is drawing a face card (king, queen or jack) from the deck of 52 cards.

Since we know that in a deck of 52 cards, there are a total 12 face cards (3 face cards each of heart, diamond, spade and club).

Number of face cards$ = 12$

Therefore, probability of getting a face card$ = \dfrac{{{\text{Number of face cards}}}}{{{\text{Total number of cards}}}} = \dfrac{{12}}{{52}} = \dfrac{3}{{13}}$.

$\left( {{\text{iii}}} \right)$ In this case, the favorable event is drawing a red face card (king, queen or jack) from the deck of 52 cards.

Since we know that in a deck of 52 cards, there are a total 6 red face cards (3 face cards of heart and 3 face cards of diamond).

Number of face cards$ = 6$

Therefore, probability of getting a red face card$ = \dfrac{{{\text{Number of red face cards}}}}{{{\text{Total number of cards}}}} = \dfrac{6}{{52}} = \dfrac{3}{{26}}$.

$\left( {{\text{iv}}} \right)$ In this case, the favorable event is drawing a jack of hearts card from the deck of 52 cards.

Since we know that in a deck of 52 cards, there is only 1 jack of hearts card.

Number of jack of hearts card$ = 1$

Therefore, probability of getting a jack of hearts card$ = \dfrac{{{\text{Number of jack of hearts card}}}}{{{\text{Total number of cards}}}} = \dfrac{1}{{52}}$.

$\left( {\text{v}} \right)$ In this case, the favorable event is drawing a spade card from the deck of 52 cards.

Since we know that in a deck of 52 cards, there are 13 spade cards.

Number of spade cards$ = 13$

Therefore, probability of getting a spade card$ = \dfrac{{{\text{Number of spade cards}}}}{{{\text{Total number of cards}}}} = \dfrac{{13}}{{52}} = \dfrac{1}{4}$.

$\left( {{\text{vi}}} \right)$ In this case, the favorable event is drawing a queen of diamonds card from the deck of 52 cards.

Since we know that in a deck of 52 cards, there is only 1 queen of diamonds card.

Number of queen of diamonds card$ = 1$

Therefore, probability of getting a queen of diamonds card$ = \dfrac{{{\text{Number of queen of diamonds card}}}}{{{\text{Total number of cards}}}} = \dfrac{1}{{52}}$.

Note- In these types of problems, we should know that in a deck of 52 cards there are 13 cards each of heart, diamond, spade and club. 13 cards of heart and 13 cards of diamond are red in colour whereas 13 cards of spade and 13 cards of club are black in colour. In these pairs of 13 cards there are 3 face cards consisting of a king, a queen and a jack.

Last updated date: 20th Sep 2023

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