Question

A card is selected from a pack of $52$ cards
(a) How many points are there in the sample space?
(b) Calculate the probability that the card is an ace of spades
(c) Calculate the probability that the card is (i) an ace (ii) black card

Answer 1

(a) When a card is selected from a pack of $52$ cards the number of possible outcomes is $52$ i.e., the sample space contains $52$ elements
Therefore there are $52$ points in the sample space
(b) Let $A$ be the event in which the card drawn is an ace of spades
Accordingly $n\left(A\right)=1$. Since there is only one ace of spade in the deck.
$\therefore P\left(A\right)=\frac{NumberofoutcomesfavourabletoA}{Totalnumberofpossibleoutcomes}=\frac{n\left(A\right)}{n\left(S\right)}=\frac{1}{52}$
(c) (i) Let $E$ be the event in which the card drawn is an ace.
Since there are $4$ aces in a pack of $52$ cards $n\left(E\right)=4$
$\therefore P\left(E\right)=\frac{NumberofoutcomesfavourabletoE}{Totalnumberofpossibleoutcomes}=\frac{n\left(E\right)}{n\left(S\right)}=\frac{4}{52}$=$\frac{1}{13}$
(ii) Let $F$ be the event in which the card drawn is black
Since there are $26$ black cards in a pack of $52$ cards $n\left(F\right)=26$
$\therefore P\left(F\right)=\frac{NumberofoutcomesfavourabletoF}{Totalnumberofpossibleoutcomes}=\frac{n\left(F\right)}{n\left(S\right)}=\frac{26}{52}=\frac{1}{2}$