Question

All kings and queens are removed from a pack of 52 cards. The remaining cards are well-shuffled and then a card is randomly drawn from it. Find the probability that this card is (a) a red face card (b) a black card.

Answer 1

Hint: To solve this problem, we need to remove all the cards given in the condition and count the cards remaining in the problem. There are two kings in red colour and there are two queens in black colour. The probability of any event is given by $\text{Probability=}\dfrac{\text{Total number of wanted outcomes}}{\text{Total number of possible outcomes}}$ .

Complete step-by-step answer:
An ideal deck of cards is a pack of 52 cards.
According to the condition we have to remove all the kings and queens.
There are 4 kings of 4 types of cards each namely king of spades, king of clubs, king of hearts and king of diamonds
Similarly, there are 4 queens of 4 types of cards each namely queen of spades, queen of clubs, queen of hearts and queen of diamonds
Therefore, in total, 8 cards are needed to be removed.
Therefore the new deck of cards is 52 – 8 = 44 cards.
Let's start by solving for the (a) part.
There were a total of 52 cards earlier with 26 red face card and 26 black face card.
Now 4 of the red cards are removed namely 2 kings of hearts and diamonds and 2 queens of hearts and diamonds respectively.
So we need to subtract these 4 cards from 26 red cards.
Therefore, the number of red cards are 26 - 4 = 22 cards
The probability is given by $\text{Probability=}\dfrac{\text{Total number of wanted outcomes}}{\text{Total number of possible outcomes}}$
Let the probability of getting a red face card be $P\left( A \right)$ .
Therefore substituting we get,
$P\left( A \right)=\dfrac{22}{44}$,
Simplifying further, we get,
$P\left( A \right)=\dfrac{22}{44}=\dfrac{1}{2}........................(i)$
Now let's get to be the (b) part.
We have to find the probability of finding the black card.
There were a total of 52 cards earlier with 26 red face card and 26 black face card.
Now 4 of the black cards are removed namely 2 kings of shapes and clubs and 2 queens of spades and clubs respectively.
So we need to subtract these 4 cards from 26 black cards.
Therefore, the number of black cards are 26 - 4 = 22 cards.
Let the probability of getting a black face card be $P\left( B \right)$ .
Therefore substituting we get,
$P\left( B \right)=\dfrac{22}{44}$,
Simplifying further, we get,
$P\left( B \right)=\dfrac{22}{44}=\dfrac{1}{2}........................(ii)$
Therefore the probability of finding a red face card is $\dfrac{1}{2}$ and the probability of finding a black card is also $\dfrac{1}{2}$.

Note: We can also find the probability of the black card by just subtracting the probability of the red card from one because there are only two possibilities in colours so other than red cards all the cards will be black face cards.
Therefore, $P\left( B \right)=1-P\left( A \right)=1-\dfrac{1}{2}=\dfrac{1}{2}$ .
We need to carefully count the cards in both cases the deck is not of 52 cards and it can be easily mistaken. The probability of any event is always less than one.