Question

One card is drawn from a well shuffled deck of cards. Find the probability of getting
(a) (i) a king of red colour (ii) a face card
     (iii) a red face card (iv) a jack of hearts
(b) (i) a spade (ii) the queen of diamonds
     (iii) neither a red card nor a queen
(c) (i) a non-face card
      (ii) a black king or a red queen

Answer 1

Hint: The deck of card has 52 cards of 4 different suits- Spades, Hearts, Clubs, Diamonds. Each suit has 13 cards. Spades and Clubs are black-coloured cards whereas Hearts and Diamonds are red coloured cards.

Complete step by step answer:
(a)
(i) As we know in the deck of cards there are a total 4 suits in which 2 are red and 2 are black. Every suit has one king card.
Hence the king of hearts and diamonds will be a red one.
Number of red colour king cards is 2
As we know probability of any event can be calculated by formula as given below:
\[\Rightarrow \text{Probability}=\dfrac{Number\text{ of favourable conditions}}{Total\text{ number of conditions}}\].
So we can calculate probability of getting a king of red colour as
\[\Rightarrow \text{Probability of getting a king of red colour}=\dfrac{Number\text{ of red colour king cards}}{Total\text{ cards}}\]
\[\Rightarrow \dfrac{2}{52}\]
\[\Rightarrow \dfrac{1}{26}\]
Hence probability of getting a king of red colour is $\dfrac{1}{26}$

(ii) The face cards are Jack, Queen and King. So we can say each suit has these three face cards. In the deck of cards there are 4 suits.
Hence, total number of face cards
\[\Rightarrow 3\times 4\]
\[\Rightarrow 12\]
\[\Rightarrow \text{Probability of getting face cards}=\dfrac{No.\text{ of face cards}}{Total\text{ cards}}\]
\[\Rightarrow \dfrac{12}{52}\]
\[\Rightarrow \dfrac{3}{13}\]
Hence probability of getting a face card is $\dfrac{3}{13}$

(iii) Red suits are hearts and diamonds. Each suit has face cards.
Hence, no. of red face cards
\[\Rightarrow 3\times 2\]
\[\Rightarrow 6\]
\[\Rightarrow \text{Probability of getting red face cards}=\dfrac{No.\text{ of red face cards}}{Total\text{ cards}}\]
\[\Rightarrow \dfrac{6}{52}\]
\[\Rightarrow \dfrac{3}{26}\]
Hence probability of getting a red face card is $\dfrac{3}{26}$

 (iv) As we know in a deck of cards each suit has one jack card. So there is only one jack of heart in a deck of cards.
\[\Rightarrow \text{Probability of getting jack of hearts}=\dfrac{No.\text{ of jack of hearts cards}}{Total\text{ cards}}\]
\[\Rightarrow \dfrac{1}{52}\]
Hence probability of getting a jack of heart is $\dfrac{1}{52}$

(b)
(i) Each suit has 13 cards.
Number of spades cards is 13
\[\Rightarrow \text{Probability of getting a spade}=\dfrac{No.\text{ of spade cards}}{Total\text{ cards}}\]
\[\Rightarrow \dfrac{13}{52}\]
\[\Rightarrow \dfrac{1}{4}\]
Hence probability of getting a spade card is $\dfrac{1}{4}$

(ii) There are four queen cards. One each in different suits. Hence, the queen of diamonds will be only one card.
\[\Rightarrow \text{Probability of getting a queen of diamonds}=\dfrac{No.\text{ of queen cards in diamond}}{Total\text{ cards}}\]
\[\Rightarrow \dfrac{1}{52}\]
Hence probability of getting a queen of diamond is $\dfrac{1}{52}$

(iii) Diamond and heart suit are red coloured cards. Total number of red cards is 26. Each suit has a queen so there are 4 queens in the deck of cards but red cards are already having 2 queens so we only have two more queen of black colour cards.
So the number of favourable outcomes of getting either a red card or a queen we will add 26 and 2.

Hence, no of cards which are either red or a queen is $26+2=28$
Therefore, no. of cards which are neither a red nor a queen is $52-28=24$
\[\Rightarrow \text{Probability of getting neither a red nor a queen}=\dfrac{24}{52}\]
\[\Rightarrow \dfrac{6}{13}\]
Hence probability of getting neither a red nor a queen is $\dfrac{6}{13}$

(c).
(i) Each suit has 3 face cards and there are 4 suits in a deck of cards.
Hence number of face cards
\[\Rightarrow 3\times 4\]
\[\Rightarrow 12\]
Number of non face cards is $52-12=40$
\[\Rightarrow \text{Probability of getting a non-face cards}=\dfrac{No.\text{ of non-face cards}}{Total\text{ cards}}\]
\[\Rightarrow \dfrac{40}{52}\]
\[\Rightarrow \dfrac{10}{13}\]
Hence probability of getting non face card is $\dfrac{10}{13}$

(ii) Spade and club are black suits. Each suit has a king.
Hence, number of black kings is 2
Heart and diamond are red suits and each suit also has a queen.
Hence number of red queens is 2
So for a number of favourable outcomes both can be possible. So for total we will add the number of black king cards and the number of red queen cards.
Number of black king and red queens is $2+2=4$
\[\Rightarrow \text{Probability of getting a black king or a red queen}=\dfrac{4}{52}\]
\[\Rightarrow \dfrac{1}{13}\]

Note: Apply the definition and formula of probability.
\[\Rightarrow \text{Probability}=\dfrac{Number\text{ of favourable conditions}}{Total\text{ number of conditions}}\]
In probability questions we have to be careful about two words ‘or’ and ‘and’.
Meaning of ‘or’ word in probability questions is to add individual possibility.
Meaning of ‘and’ word in probability questions is to multiply individual possibilities.