Question

One card is randomly drawn from a pack of 52 cards. If drawing a spade card is denoted by event A and drawing a card from ace to ten (non-face card) is denoted by B then write the sets showing the following events:

(1) U
(2 ) A
(3 ) B
(4) A ∪ B
(5) A ∩ B
(6) B’

Answer 1

A pack of 52 cards consists of 13 spade cards, 13 club cards, 13 heart cards and 13 diamond cards.

Let, S = Spade card,
C = Club card,
H = Heart card,
D = Diamond card,
J = Jack card,
Q = Queen card,
K = King card,
A = Ace card

Non-face card: Spade: S₂, S₃, S₄, …, S₁₀
Club: C₂, C₃, C₄, …, C₁₀
Heart: H₂, H₃, H₄, …, H₁₀
Diamond: D₂, D₃, D₄, …, D₁₀

Total number of ways of drawing a card randomly from a pack of 52 cards is ⁵²C₂ = 52

(1) U: The sample space of drawing a card randomly from a pack of 52 cards is expressed as follows:

u = {S_A, S₂, S₃, S₄, …….. S₁₀, S_J, S_Q, S_K
C_A, C₂, C₃, C₄, …….. C₁₀, C_J, C_Q, C_K
H_A, H₂, H₃, H₄, …….. H₁₀, H_J, H_Q, H_K
D_A, D₂, D₃, D₄, …….. D₁₀, D_J, D_Q, D_K}

(2) Event A = Drawing a spade card
S_A, S₂, S₃, S₄, …….. S₁₀, S_J, S_Q, S_K

(3) Event B = Drawing a card from ace to ten
= {S_A, S₂, S₃, …….. S₁₀, C_A, C₂, C₃, …….. C₁₀, H_A, H₂, H₃, …….. H₁₀, D_A, D₂, D₃, …….. D₁₀,}

(4) Event A ∪ B = Drawing a spade card or a card from ace to ten
= {S_A, S₂, S₃, S₄, …….. S₁₀, S_J, S_Q, S_K, C_A, C₂, C₃, …….. C₁₀, H_A, H₂, H₃, …….. H₁₀, D_A, D₂, D₃, …….. D₁₀}

(5 ) Event A ∩ B = Drawing a spade card and from ace to ten
= {S_A, S₂, S₃, …….. S₁₀}

(6) Event B’ = Drawing a face card (or not from ace to ten)
= {S_J, S_Q, S_K, C_J, C_Q, C_K, H_J, H_Q, H_K, D_J, D_Q, D_K}