Question

A single card is drawn from a standard 52-card deck.Find the conditional probability that
a)THe card is red,given that it is a face card
b)The card is a heart, given that it is an ace
c)The card is a face card,given that it is red
d)The card is black, given that it is a club
e)The card is a jack,given that it is red
f)The card is a club,given that it is black

Answer 1

total cards in a deck = 52
there are 4 sets of cards(13 in each set)
hearts and diamonds are red
clubs and spades are black

a) The card is red,given that it is a face card
E = card is red
F = card is a face card
EΠF = card which is a red face card

n(EΠF) = 6
P(EΠF) = 6/52 = 3/26
n(F) = 12 (=3 in each set × 4)
P(F) = 12/52 = 3/13

P(E|F) = P(EΠF) / P(F) = 3/26 × 13/3 = 1/2

b)The card is a heart, given that it is an ace
E = card is a heart
F = card is an ace
EΠF = the card ace of heart

n(EΠF) = 1 (only one ace of heart)
P(EΠF) = 1/52
n(F) = 4 (=1 in each set × 4)
P(F) =4/52 = 1/13

P(E|F) = P(EΠF) / P(F) = 1/52 × 13/1 = 1/4

c)The card is a face card,given that it is red
E = card is a face card
F = card is red
EΠF = the card is a red face card

n(EΠF) = 6 (3 in each set × 2 red sets)
P(EΠF) = 6/52 = 3/26
n(F) = 26 
P(F) =26/52 = 1/2

P(E|F) = P(EΠF) / P(F) = 3/26 × 2/1 = 3/13

d)The card is black, given that it is a club
E = card is black
F = card is a club
EΠF = the card is a black club card

n(EΠF) = 13 (all 13 club cards are black)
P(EΠF) = 13/52 = 1/4
n(F) = 13 
P(F) =13/52 = 1/4

P(E|F) = P(EΠF) / P(F) = 1/4 × 4/1 = 1

e)The card is a jack,given that it is red
E = card is a jack
F = card is red
EΠF = the card is a red jack card

n(EΠF) = 2 (1 in heart and 1 in diamond)
P(EΠF) = 2/52 = 1/26
n(F) = 26
P(F) =26/52 = 1/2

P(E|F) = P(EΠF) / P(F) = 1/26 × 2/1 = 1/13

f)The card is a club,given that it is black
E = card is club
F = card is a black
EΠF = the card is a black club card

n(EΠF) = 13 (all 13 club cards are black)
P(EΠF) = 13/52 = 1/4
n(F) = 26
P(F) =26/52 = 1/2

P(E|F) = P(EΠF) / P(F) = 1/4 × 2/1 = 1/2

Answer 2

P( B | A) = probability of occurrence of event B, given that event A has occurred
        = P(A П B) / P(A)    = conditional probability
              A Π B  means the intersection of events A and B, occurring together.

a)   Face cards = {J, Q, K } ie., 3 of each suite.          So Prob = 12/52 = 3/13
Num of red cards = 26.     Red cards П face cards = 6        So prob = 6/52 = 3/26
                     Prob = (3/26) / (3/13) = 1/2
===
b)    Hearts AND aces = 1            number of Aces = 4
                 Prob = (1/52) / (4/52) = 1/4
===
c) Face cards AND Red cards = 6              number of red cards = 26
                   Prob = (6/52 ) / (26/52) = 3/13
===
d)    Number of Black cards П Clubs = 13          Number of club cards = 13
               Prob = (13/52) / (13/52) = 1
===
e) Number of Jacks П Reds = 2                  Number of red cards = 26
                       prob = (2/52) / (26/52) = 1/13
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f)    Number of clubs П Black cards = 13          number of black cards = 26
                   Prob = (13/52) / (26/52) = 1/2