Question

# All the red face cards are removed from a pack of 52 playing cards. A card is drawn at random from the remaining cards, after reshuffling them. Find the probability that the drawn card is(a) of red color(b) a queen(c) an ace(d) a face card

Hint: A standard deck of playing cards has 4 suits of 13 each. And each suit has 3 face cards (a king, a queen, and a jack) so in total 12 cards are present in a standard deck.

As we know a standard deck of playing cards has 52 cards in it broken into 4 suits of 13 cards each.
Each suit has 3 face cards (a king, a queen, and a jack) so, total cards are 12. As, there are two colours present in 52 cards i.e. Red and Black.
So, red cards should be â€˜6â€™ i.e. half of total face cards.
Hence, total cards remained in deck after removing all red face cards will be 46 i.e.,
52-6 = 46
So, total cards present in the deck = 46.
Now, we need to find probability when a card is drawn at random from the remaining cards and the drawn card is:
(a) Of red colour
Total red colour cards in 52 cards are = 26(13 for diamond + 13 for hearts)
Remove cards of red colour = 6
Hence, remaining cards of red colour = 26-6 = 20
As, probability of any event can be given as
$P(E)=\dfrac{\text{Number of favourable cases}}{\text{Total cases}}$
Hence, probability of drawing a red colour card be
$P=\dfrac{20}{46}=\dfrac{10}{23}$
(b) A queen
Total number of queens present in 52 cards = 4(one of each suit).
As queen is a face card, so there will be only 2 queens in the remaining 46 cards as 2 queens (hearts and diamonds) are already removed with the 6 red face cards.
Hence, queens present in remaining cards = 2
So, probability of drawing a queen can be given as
$P=\dfrac{2}{46}=\dfrac{1}{23}$
(c) An ace
Total number of aces in 52 cards = 4 (one for each suit)
As ace is not a face card. So, all the 4 aces will be present in 46 remaining cards after removing all red face cards.
Hence, probability of drawing an ace, we get
$P=\dfrac{4}{46}=\dfrac{2}{23}$
(d) A face card
Total number of face cards present in 52 cards be
= 12 (3 of each suit) or
= 12 (6 of each colour)
Now, when we remove 6 red face cards, then remaining face cards would be 6 each of black colour.
Hence, the total face cards present in 46 cards is 6.
So, probability of drawing face cards from remaining cards will be,
$P=\dfrac{6}{46}=\dfrac{3}{23}$

Note: One can include â€˜Aceâ€™ as a face card as well and assume total face cards would be 16 which is wrong. It is a general confusion with students. Hence, face cards include only queen, jack and king and in total 12 cards are there.
All 13 cards include face cards i.e. 1 king, 1 queen, 1 jack and cards numbered from 1,2,3â€¦â€¦..10, where 1 is termed as an Ace.