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19 December, 01:31

A single card is drawn from a deck of 52 cards, find the following

probabilities:

a. result is a queen

b. result is a queen result is a face card

c. result is a heart

d. result is a heart | result is a red card

e. result is a heart result is a black card

f. result is a heart result is not a diamond

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Answers (1)
  1. 19 December, 03:22
    0
    a. 1/13

    b. 1/78

    c. 1/4

    d. 1/12

    e. 1/6

    f. 1/4

    Step-by-step explanation:

    First, lets write some basic data

    There are total 52 cards. 26 cards are red and 26 cards are black Out of 26 red cards; 13 cards are of diamond and 13 cards are of heart Out of 26 black cards; 13 cards are of spade and 13 cards are of clubs In spade, clubs, heart and diamond there is one card of ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, King, Queen, Jack

    a. result is a queen

    Each set has one queen; 1 of spade, 1 of clubs, 1 of heart, 1 of heart

    There are total 4 queens

    Total number of cards is 52

    Probability = Number of favourable outcomes/total number of outcomes

    Probability = 4/52

    Probability = 1/13

    b. result is a queen result is a face card

    From my understanding, this question is result is a queen given that result is a face card.

    Queen: 4 cards

    Face card means King, Queen and Jack

    Each set has one King, one Queen and one Jack

    There are total 4 sets

    Therefore, there are 4 Kings, 4 Queens and 4 Jacks

    Face cards: 12-4 (because 4 queens have already been taken out) = 8

    Total number of cards: 52-4 (because 4 queens have already been taken out) = 48

    Probability (A|B) = P (A) * P (A|B)

    P (A) = result is a queen = 4/52 = 1/13

    P (A|B) = result is a face card = 8/48 = 1/6

    Probability (A|B) = P (A) * P (A|B)

    Probability (A|B) = 1/13 * 1/6

    Probability (A|B) = 1/78 or 0.01282

    c. result is a heart

    Out of the 52 cards, there are 26 red cards and out of the 26 red cards, there are 13 cards of heart

    Heart = 13 cards

    Total number of cards = 52 cards

    Probability = Number of favourable outcomes/total number of outcomes

    Probability = 13/52

    Probability = 1/4

    d. result is a heart | result is a red card

    Result is a heart = P (A) = 13/52 = 1/4

    Result is a red card=26/52 Total number of red cards is 26 but 13 cards (heart) are already taken out so 13 will be subtracted from the numerator and denominator

    = 13/39 = 1/3

    Result is a heart given that result is a red card P (A|B) = 1/3

    Probability (A|B) = P (A) * P (A|B)

    Probability (A|B) = 1/4 * 1/3

    Probability (A|B) = 1/12

    e. result is a heart result is a black card

    Result is a heart = P (A) = 13/52 = 1/4

    Result is a black card=26/52 Total number of cards is 52 but 13 cards (heart) are already taken out so 13 will be subtracted from the denominator

    = 26/39 = 2/3

    Result is a heart given that result is a black card P (A|B) = 2/3

    Probability (A|B) = P (A) * P (A|B)

    Probability (A|B) = 1/4 * 2/3 = 1/6

    f. result is a heart result is not a diamond

    Result is a heart = P (A) = 13/52 = 1/4

    Result is not a diamond = there are 13 diamonds in a deck so cards that are not diamond are 52-13 = 39

    Total cards = 52 - 13 (because hearts are already taken out) = 39

    = 39/39 = 1

    Probability (A|B) = P (A) * P (A|B)

    Probability (A|B) = 1/4 * 1 = 1/4

    !
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