Question

Find the probability of drawing an ace or a spade from a deck of $52$ cards in a single draw?

Answer 1

Find the Probability of drawing an ace or a spade card.

There are total $52$ cards in a packet.

Also, there are $4$ ace and $13$ spade cards.

So that the probability of drawing an ace card is :

$P\left(A\right)=\frac{4}{52}=\frac{1}{13}$

and

The probability of drawing a spade card is:-

$P\left(S\right)=\frac{13}{52}=\frac{1}{4}$

Also there is one card that is both a spade and an ace. So the probability of that is $P(A\cap S)=\frac{1}{52}$. This probability should be subtracted from the total probability so that the card should not be counted twice.

So the required probability of drawing an ace or spade card is :

$$\begin{gathered} P(aceorspade)=P\left(A\right)+P\left(S\right)-P(A\cap S) \\ \Rightarrow P(aceorspade)=\frac{1}{13}+\frac{1}{4}-\frac{1}{52} \\ \Rightarrow P(aceorspade)=\frac{4+13-1}{52} \\ \Rightarrow P(aceorspade)=\frac{16}{52} \\ \Rightarrow P(aceorspade)=\frac{4}{13} \end{gathered}$$

Therefor the required probability is $\frac{\mathbf{4}}{\mathbf{13}}$.