Question

A card drawn at random from a well shuffled pack of 52 playing cards. What is the probability of getting neither a red card nor a queen?
A) $ \dfrac{5}{{12}} $
B) $ \dfrac{3}{{13}} $
C) $ \dfrac{2}{{11}} $
D) $ \dfrac{6}{{13}} $

Answer 1

Hint: The number of choosing the red cards and queens can be subtracted from the total number of cards, it will become the favorable outcome as we have to calculate the probability to get either red or queen cards.
Formula to calculate probability:
 $ P = \dfrac{f}{T} $ where,
P = Probability
F = Favorable outcomes
T = Total outcomes

Complete step-by-step answer:
In a deck of 52 cards, the number of red cards and queens are as follows:
Half the cards are red and half the cards are black, so
The number of red cards = $ \dfrac{{52}}{2} $
                                              = 26
Number of queens = 4 , out of which 2 are black and 2 are red.
Out of 52 cards, if we don’t want red cards:
52 – 26
We also don’t want queens, but 2 out of 4 queens are red and are counted in 26 red cards, so we have to delete 2 black queens more:
52 – 26 – 2
52 – 28 = 24 ____________ (1)
Calculating the required probability:
 $ \Rightarrow $ Favorable outcomes (f) = Total cards – red and queen cards
                                      = 24 [from(1)]
 $ \Rightarrow $ Total outcomes (T) = Total number of cards
                             = 52
Applying the formula of probability:
 $ \Rightarrow P = \dfrac{f}{T} $
Substituting the values, we get:
 $
 \Rightarrow P = \dfrac{{24}}{{52}} \\
\Rightarrow P = \dfrac{6}{{13}} \\
  $
Therefore, if a card is drawn at random from a well shuffled pack of 52 playing cards, the probability of getting neither a red card nor a queen is $ \dfrac{6}{{13}} $ and the correct option is D.

So, the correct answer is “Option D”.

Note: Queen is called a face card and in total, there are 3 face cards, 3 X 4 =12 in number as each face card is present in each suit (4 suits).
Out of 4, 2 are present as red and 2 as black.
Basics of cards to remember:
Total cards = 52
Black cards = 26
Red cards = 26
Suits = 4 (Club, Heart, Diamond and Spade)
Face cards = 12 (3 X 4)