Question

A card is drawn from a well shuffled pack of 52 cards. Find the probability of it being a heart or a queen.

Answer 1

One card can be drawn from the pack of 52 cards in ⁵²C₁ = 52 ways
∴ n(S) = 52
Also, the pack of 52 cards consists of 13 heart cards and 4 queen cards
Let A be the event that a card drawn is a heart.
A heart card can be drawn from 13 heart cards in ¹³C₁ ways
∴ n(A) = `""^13"C"_1`

∴ P(A) = `("n"("A"))/("n"("S")) = (""^13"C"_1)/52= 13/52`
Let B be the event that a card drawn is queen.
A queen card can be drawn from 4 queen cards in ⁴C₁ ways
∴ n(B) = `""^4"C"_1`

∴ P(B) = `("n"("B"))/("n"("S")) = (""^4"C"_1)/52 = 4/52`
There is one queen card out of 4 which is also a heart card
∴ n(A ∩ B) = `""^1"C"_1`

∴ P(A ∩ B)  = `("n"("A" ∩ "B"))/("n"("S")) = (""^1"C"_1)/52 = 1/52`
∴ P(card is a heart or a queen)
= P(A ∪ B)
= P(A) + P(B) – P(A ∩ B)

= `13/52+4/52-1/52`

= `(13 + 4 - 1)/52`

= `16/52`

∴ P(A ∪ B) = `4/13`