Probability of card (popular questions)

1. Deck of Cards Questions
- There are 52 cards in a standard deck of cards
- There are 4 of each card (4 Aces, 4 Kings, 4 Queens, etc.)
- There are 4 suits (Clubs, Hearts, Diamonds, and Spades) and there are 13 cards in each suit
(Clubs/Spades are black, Hearts/Diamonds are red)
- Without replacement means the card IS NOT put back into the deck. With replacement means the card IS
put back into the deck.
What is the probability of picking up an ace in a 52 card deck?
The probability of picking up an ace in a 52 deck of cards is 4/52
What is the probability of picking up an ace or king in a 52 card deck?
The probability of picking up an ace or a king is 8/52
What is the probability that when two cards are drawn from a deck of cards without replacement
that both of them will be 8’s?
𝑃(𝐵𝑜𝑡ℎ 𝑎𝑟𝑒 8 ′ 𝑠) = 𝑃(𝐹𝑖𝑟𝑠𝑡 𝑐𝑎𝑟𝑑 𝑖𝑠 𝑎𝑛 8) ∙ 𝑃(𝑆𝑒𝑐𝑜𝑛𝑑 𝑐𝑎𝑟𝑑 𝑖𝑠 𝑎𝑛 8)
𝑃(𝐹𝑖𝑟𝑠𝑡 𝑐𝑎𝑟𝑑 𝑖𝑠 𝑎𝑛 8) = 4/52
There are three 8’s left in the deck if one is pulled and not replaced, and 51 total cards remaining.
𝑃(𝑆𝑒𝑐𝑜𝑛𝑑 𝑐𝑎𝑟𝑑 𝑖𝑠 𝑎𝑛 8) = 3/51
𝑃(𝐵𝑜𝑡ℎ 𝑎𝑟𝑒 8 ′ 𝑠) = 4/ 52 ∙ 3/ 51 = 12 /2652 = 1 /221 = .0045 𝑜𝑟 .45%
What is the probability that both cards drawn (without replacement) will be spades?
𝑃(𝐵𝑜𝑡ℎ 𝑎𝑟𝑒 𝑠𝑝𝑎𝑑𝑒𝑠) = 𝑃(𝐹𝑖𝑟𝑠𝑡 𝑐𝑎𝑟𝑑 𝑖𝑠 𝑎 𝑠𝑝𝑎𝑑𝑒) ∙ 𝑃(𝑆𝑒𝑐𝑜𝑛𝑑 𝑐𝑎𝑟𝑑 𝑖𝑠 𝑎 𝑠𝑝𝑎𝑑𝑒)
𝑃(𝐹𝑖𝑟𝑠𝑡 𝑐𝑎𝑟𝑑 𝑖𝑠 𝑎 𝑠𝑝𝑎𝑑𝑒) = 13 /52
There are 12 spades left in the deck if one is pulled and not replaced, and 51 total cards remaining.

2. 𝑃(𝑆𝑒𝑐𝑜𝑛𝑑 𝑐𝑎𝑟𝑑 𝑖𝑠 𝑎 𝑠𝑝𝑎𝑑𝑒) = 12 /51
𝑃(𝐵𝑜𝑡ℎ 𝑎𝑟𝑒 𝑠𝑝𝑎𝑑e) = 13/52 ∙ 12 /51 = 156 /2652 = 1 /17 = .0588 𝑜𝑟 5.88%
What is the probability of drawing a red king and then a black 7 without replacement?
𝑃(𝑅𝑒𝑑 𝑘𝑖𝑛𝑔 𝑡ℎ𝑒𝑛 𝑏𝑙𝑎𝑐𝑘 7) = 𝑃(𝑅𝑒𝑑 𝑘𝑖𝑛𝑔) ∙ 𝑃(𝑆𝑒𝑐𝑜𝑛𝑑 𝑐𝑎𝑟𝑑 𝑖𝑠 𝑎 𝑏𝑙𝑎𝑐𝑘 𝑠𝑒𝑣𝑒𝑛)
There are 4 of each card, so there are 2 red and 2 black of each card. This means we have 2 red kings in
the deck, and 2 black 7’s in the deck.
𝑃(𝑅𝑒𝑑 𝑘𝑖𝑛𝑔) = 2 /52 *Even with a red king drawn first, there will still be 2 black 7’s in the deck, but only 51
cards remaining.
𝑃(𝑆𝑒𝑐𝑜𝑛𝑑 𝑐𝑎𝑟𝑑 𝑖𝑠 𝑎 𝑏𝑙𝑎𝑐𝑘 𝑠𝑒𝑣𝑒𝑛) = 2/ 51
𝑃(𝑅𝑒𝑑 𝑘𝑖𝑛𝑔 𝑡ℎ𝑒𝑛 𝑏𝑙𝑎𝑐𝑘 7) = 2 /52 ∙ 2/ 51 = 4 /2652 = 1 /663 = .0015 𝑜𝑟 .15%
What is the probability of drawing two face cards, and then 2 numbered cards, without
replacement?
There are 12 face cards (Kings, queens, and jacks) and there are 36 numbered cards.
After the first face card is drawn, there will be 11 face cards leftover, and 51 total cards remaining.
𝑃(2 𝑓𝑎𝑐𝑒 𝑐𝑎𝑟𝑑𝑠) = 12/ 52 ∙ 11/ 51
Now we only have 50 cards left in the deck, but all 36 of the numbered cards are still in there. After one is
drawn, there are 35 numbered cards remaining of the 49 total cards that now remain.
𝑃(2 𝑛𝑢𝑚𝑏𝑒𝑟𝑒𝑑 𝑐𝑎𝑟𝑑𝑠) = 36 /50 ∙ 35 /49
𝑃(2 𝑓𝑎𝑐𝑒 𝑐𝑎𝑟𝑑𝑠 𝑡ℎ𝑒𝑛 2 # 𝑐𝑎𝑟𝑑𝑠) = 𝑃(2 𝑓𝑎𝑐𝑒 𝑐𝑎𝑟𝑑𝑠) ∙ 𝑃(2 𝑛𝑢𝑚𝑏𝑒𝑟𝑒𝑑 𝑐𝑎𝑟𝑑𝑠)
𝑃(2 𝑓𝑎𝑐𝑒 𝑐𝑎𝑟𝑑𝑠 𝑡ℎ𝑒𝑛 2 # 𝑐𝑎𝑟𝑑𝑠) = 12 /52 ∙ 11 /51 ∙ 36 /50 ∙ 35/ 49 = 166320/ 6497400 = 198/ 7735 = .0256
𝑜𝑟 2.56%
What is the probability of drawing an Ace 3 times in a row with replacement?
𝑃(3 𝐴𝑐𝑒𝑠 𝑖𝑛 𝑎 𝑟𝑜𝑤) = 𝑃(𝐶𝑎𝑟𝑑 1 𝑖𝑠 𝑎𝑛 𝐴𝑐𝑒) ∙ 𝑃(𝐶𝑎𝑟𝑑 2 𝑖𝑠 𝑎𝑛 𝐴𝑐𝑒) ∙ 𝑃(𝐶𝑎𝑟𝑑 3 𝑖𝑠 𝑎𝑛 𝐴𝑐𝑒)
This time, we are replacing the card, which means there will always be 4 Aces in the deck, and always 52
total cards. 𝑃(𝐺𝑒𝑡𝑡𝑖𝑛𝑔 𝑎𝑛 𝐴𝑐𝑒) = 4 /52
𝑃(3 𝐴𝑐𝑒𝑠 𝑖𝑛 𝑎 𝑟𝑜𝑤) = 4 /52 ∙ 4/ 52 ∙ 4/ 52 = 1 /2197 = .000455 = .0455%
A face card is drawn.
12/52 = 3/13
A red card or a card showing a 5 is drawn.
P(Red OR 5) = P(Red) + P(5) - P(Red AND 5) = 26/52 + 4/52 - 2/52 = 28/52 = 7/13
A non-face card or a 7 is drawn.
P(non-face OR 7) = P(non-face) + P(7) - P(non-face AND 7) = 40/52 + 4/52 - 4/52 = 40/52 = 10/13
A card drawn is neither a king nor a spade.

3. There are 4 kings and there are 13 spades, with one that is both a king and a spade, so there are 4 + 13 - 1
= 16 cards that are either kings or spades. That leaves 52-16=36 cards that are neither a king nor a spade.
Our desired probability is 36/52 = 9/13.
A card that is a black face card is drawn.
There are 6 black face cards, so the probability is 6/52 = 3/26.
A card that is not a face card is drawn.
We know from question (1) that P(face card) = 3/13. Because the event "non-face card" is the complement
to "face card," P(non-face card) = 1 - P(face card) = 1 - 3/13 = 10/13
If you draw 3 cards from a deck one at a time what is the probability:
All 3 cards are Red?
P(1st is red ∩ 2nd is red ∩ 3rd is red)
= P(1st is red)*P(2nd is red)*P(3rd is red) by independence = (26/52) * (25/51) * (24/50) = .1176
The king, queen and jack of clubs are removed from a deck of 52 playing cards and then shuffled. A
card is drawn from the remaining cards. Find the probability of getting:
(i) a heart
(ii) a queen
(iii) a club
(iv) ‘9’ of red color
Solution:
Total number of card in a deck = 52
Card removed king, queen and jack of clubs
Therefore, remaining cards = 52 - 3=49
(i) a heart
Number of hearts in a deck of 52 cards= 13
= 13/49
(ii) a queen
Number of queen = 3
[Since club’s queen is already removed]
= 3/49
(iii) a club
Number of clubs in a deck in a deck of 52 cards = 13
According to the question, the king, queen and jack of clubs are removed from a deck of 52 playing cards In
this case, total number of clubs = 13 - 3 = 10
= 10/49
(iv) ‘9’ of red color
Cards of hearts and diamonds are red cards
The card 9 in each suit, hearts and diamonds = 1

4. Therefore, total number of ‘9’ of red color = 2
= 2/49
All kings, jacks, diamonds have been removed from a pack of 52 playing cards and the remaining
cards are well shuffled. A card is drawn from the remaining pack. Find the probability that the card
drawn is:
(i) a red queen
(ii) a face card
(iii) a black card
(iv) a hear
Solution:
Number of kings in a deck 52 cards = 4
Number of jacks in a deck 52 cards = 4
Number of diamonds in a deck 52 cards = 13
Total number of cards removed = (4 kings + 4 jacks + 11 diamonds) = 19 cards
[Excluding the diamond king and jack there are 11 diamonds]
Total number of cards after removing all kings, jacks, diamonds = 52 - 19 = 33
(i) a red queen
Queen of heart and queen of diamond are two red queens
Queen of diamond is already removed.
So, there is 1 red queen out of 33 cards
Therefore, the probability of getting ‘a red queen’
= 1/33
(ii) a face card
Number of face cards after removing all kings, jacks, diamonds = 3
Therefore, the probability of getting ‘a face card’
= 3/33 = 1/11
(iii) a black card
Cards of spades and clubs are black cards.
Number of spades = 13 - 2 = 11, since king and jack are removed
Number of clubs = 13 - 2 = 11, since king and jack are removed
Therefore, in this case, total number of black cards = 11 + 11 = 22
Therefore, the probability of getting ‘a black card’
= 22/33 = 2/3

5. (iv) a heart
Number of hearts = 13
Therefore, in this case, total number of hearts = 13 - 2 = 11, since king and jack are removed
= 11/33 = 1/3
A card is drawn from a well-shuffled pack of 52 cards. Find the probability that the card drawn is:
(i) a red face card
(ii) neither a club nor a spade
(iii) neither an ace nor a king of red color
(iv) neither a red card nor a queen
(v) neither a red card nor a black king.
Solution:
Total number of card in a pack of well-shuffled cards = 52
(i) a red face card
Cards of hearts and diamonds are red cards.
Number of face card in hearts = 3
Number of face card in diamonds = 3
Total number of red face card out of 52 cards = 3 + 3 = 6
Therefore, the probability of getting ‘a red face card’
= 6/52 = 3/26
(ii) neither a club nor a spade
Number of clubs = 13
Number of spades = 13
Number of club and spade = 13 + 13 = 26
Number of card which is neither a club nor a spade = 52 - 26 = 26
= 26/52 = 1/2
(iii) neither an ace nor a king of red color
Number of ace in a deck 52 cards = 4
Number of king of red color in a deck 52 cards = (1 diamond king + 1 heart king) = 2
Number of ace and king of red color = 4 + 2 = 6
Number of card which is neither an ace nor a king of red color = 52 - 6 = 46
= 46/52 = 23/26
(iv) neither a red card nor a queen
Number of hearts in a deck 52 cards = 13
Number of diamonds in a deck 52 cards = 13
Number of queen in a deck 52 cards = 4

6. Total number of red card and queen = 13 + 13 + 2 = 28,
[since queen of heart and queen of diamond are removed]
Number of card which is neither a red card nor a queen = 52 - 28 = 24
= 24/52 = 6/13
(v) neither a red card nor a black king.
Number of hearts in a deck 52 cards = 13
Number of diamonds in a deck 52 cards = 13
Number of black king in a deck 52 cards = (1 king of spade + 1 king of club) = 2
Total number of red card and black king = 13 + 13 + 2 = 28
Number of card which is neither a red card nor a black king = 52 - 28 = 24
= 24/5 = 6/13