Question

One card is drawn from well shuffled deck of 52 cards. Find the probability of getting:
(i) A king of red colour,
(ii) A face card,
(iii) The jack of hearts,
(iv) A red face card,
(v) A spade,
(vi) The queen of diamonds.

Answer 1

Given that, one card is drawn from a well shuffled deck of $52$ cards.

To find out: The probability of getting:

(i) A king of red colour,

(ii) A face card,

(iii) The jack of hearts,

(iv) A red face card,

(v) A spade,

(vi) The queen of diamonds.

We know that,

The probability of an event $E,P\left(E\right)=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}=\frac{n\left(E\right)}{n\left(S\right)}$

Applying the concept of probability as shown above, let's calculate all the probabilities one by one.

$\left(i\right)$ Probability of getting a king of red colour:

We know that, there are $26$ red cards, $13$ each of hearts and diamonds. Hence, there will be $1$ king each in hearts and diamonds.

$\therefore$ Number of kings of red colour, $n\left(E\right)=2$

Also, total number of cards, $n\left(S\right)=52$

$\therefore$ Probability of getting a king of red colour $=\frac{n\left(E\right)}{n\left(S\right)}=\frac{2}{52}$

$\Rightarrow \frac{1}{26}$

Hence, the required probability is $\frac{1}{26}$.

$\left(ii\right)$ Probability of getting a face card:

We know that, each suit has $3$ face cards (Jack, Queen and King). Hence, there will be $12$ face cards in total.

$\therefore$ Number of face cards, $n\left(E\right)=12$

Also, total number of cards, $n\left(S\right)=52$

$\therefore$ Probability of getting a face card $=\frac{n\left(E\right)}{n\left(S\right)}=\frac{12}{52}$

$\Rightarrow \frac{3}{13}$

Hence, the required probability is $\frac{3}{13}$.

$\left(iii\right)$ Probability of getting the jack of hearts:

We know that, there is only one jack in hearts.

$\therefore$ Number of jack of hearts, $n\left(E\right)=1$

Also, total number of cards, $n\left(S\right)=52$

$\therefore$ Probability of getting the jack of hearts $=\frac{n\left(E\right)}{n\left(S\right)}=\frac{1}{52}$

Hence, the required probability is $\frac{1}{52}$.

$\left(iv\right)$ Probability of getting a red face card:

We know that, there are $26$ red cards, $13$ each of hearts and diamonds. Both hearts and diamonds have $3$ face cards each.

$\therefore$ Number of red face cards, $n\left(E\right)=6$

Also, total number of cards, $n\left(S\right)=52$

$\therefore$ Probability of getting a red face card $=\frac{n\left(E\right)}{n\left(S\right)}=\frac{6}{52}$

$\Rightarrow \frac{3}{26}$

Hence, the required probability is $\frac{3}{26}$.

$\left(v\right)$ Probability of getting a spade:

We know that, there are $13$ cards of spades.

$\therefore$ Number of spades, $n\left(E\right)=13$

Also, total number of cards, $n\left(S\right)=52$

$\therefore$ Probability of getting a spade $=\frac{n\left(E\right)}{n\left(S\right)}=\frac{13}{52}$

$\Rightarrow \frac{1}{4}$

Hence, the required probability is $\frac{1}{4}$.

$\left(vi\right)$ Probability of getting the queen of diamonds:

We know that, there are $13$ cards of diamonds. There is $1$ queen of diamonds.

$\therefore$ Number of queens in diamonds, $n\left(E\right)=1$

Also, total number of cards, $n\left(S\right)=52$

$\therefore$ Probability of getting the queen of diamonds $=\frac{n\left(E\right)}{n\left(S\right)}=\frac{1}{52}$

Hence, the required probability is $\frac{1}{52}$.