Question

Let A denote the event of choosing a black spade card at random from a standard 52-card playing card set with red and black colors. Also, let B denote the event of choosing a royal card, i.e., a jack, queen, or king at random from a similar playing card set.

1. Write out the elements in the sets (i) A, (ii) B, (iii) A ∩ B, and (iv) A ∪ B using correct set notation.
2. For each of the sets in the previous part, calculate the associated probabilities using the number of elements in the set out of 52 total cards.

3. Verify Equation (4.5) is correct for A ∪ B above.

Answer 1

1.

(A)

A=black spade card.

A={ACE,2,3,4,5,6,7,8,9,10,JACK,QUEEN.KING}=13 elements

(B)

]B=a royal card

black card set:=6 elements

Spade=S=3 royal cards,Club=C=3 royal cards

red card set:=6 elements

Heart=H=3 royal cards,Diamond=D=3 royal cards

total=12

B={JACK,QUEEN,KING,JACK,QUEEN,KING,JACK,QUEEN,KING,JACK,QUEEN,KING}=12 elements.

(C)

A ∩ B=a royal card of spade

A ∩ B={JACK,QUEEN,KING}=3 ELEMENTS

(D)

A ∪ B={a spade card or a royal card}=13 spade card and 9 royal cards(from diamond,heart,club)

A ∪ B={ACE,2,3,4,5,6,7,8,9,10,JACK,QUEEN.KING,,JACK,QUEEN,KING,JACK,QUEEN,KING,JACK,QUEEN,KING}=22

2.

P(A)==0.25

P(B)==0.2307

P(A ∩ B)==0.0576

P(A ∪ B)==0.423 --------------------(1)

3.

We know,

P(A ∪ B)=0.423 (from equation 1)

P(A ∪ B)=P(A)+P(B)-P(A ∩ B)

P(A ∪ B)=0.25+0.2307-0.0576

P(A ∪ B)=0.423 (which is equal to the value of P(A ∪ B) in equation 1)

hence verified.

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