Question

A card is drawn at random from a well shuffled pack of 52 cards.Find the probability that the card drawn is :

(ii) a spade or a face card.

Answer 1

Total number of cards = 52
∴ n (S) = 52
Let C be the event of drawing a spade card and D be the event of drawing a face card.
We know:
Number of spade cards = 13
$\therefore$ n (C) = 13
Number of face cards = 12
$\therefore$ n (D) = 12
And number of face cards of spade = 3
$\therefore$ n (C $\cap$ D) = 3
P (C) = $\frac{n\left(C\right)}{n\left(S\right)}=\frac{13}{52}=\frac{1}{4}$

P (D) = $\frac{n\left(D\right)}{n\left(S\right)}=\frac{12}{52}=\frac{3}{13}$

And,
P (C $\cap$ D) = $\frac{n(C\cap D)}{n\left(S\right)}=\frac{3}{52}$

Now, on applying the addition theorem of probability, we get:
P (C $\cup$ D) = P (C) + P (D) $-$ P (C $\cap$ D)

= $\frac{1}{4}+\frac{3}{13}-\frac{3}{52}=\frac{11}{26}$

$\therefore$ Probability of drawing a spade card or a face card = $\frac{11}{26}$