Question

Question: What is the probability that a card selected at random from a standard deck of 52 cards is an ace or a heart?

Short answer

Answer

The probability that a card selected at random from a standard deck of 52 cards is an ace or a heart is .

Step-by-step solution

 Given  

A standard deck of 52 cards.

The Concept of probability

If represents the sample space and $\mathit{E}$represents the event. Then the probability of occurrence of favourable event is given by the formula as below:

$\mathit{P}\mathbf{\left(}\mathit{E}\mathbf{\right)}\mathbf{=}\frac{\mathbf{n}\mathbf{\left(}\mathbf{E}\mathbf{\right)}}{\mathbf{n}\mathbf{\left(}\mathbf{S}\mathbf{\right)}}$

Determine the probability

As per the problem we have been asked to find the probability that a card selected at random from a standard deck of 52 cards is an ace or a heart.

If $S$represents the sample space and $E$represents the event. Then the probability of occurrence of favourable event is given by the formula as below:

$P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}.$

As we know that there are 4 aces (A), 13 heart (H) and 1 ace of heart so we can represent it as below:

$n\left(A\right)=4,n\left(H\right)=13,n(A\cap H)=1$

Now using the formula

$$\begin{gathered} n\left(E\right)=n(A\cup B) \\ =n\left(A\right)+n\left(B\right)-n(A\cap B) \\ n\left(E\right)=4+13-1 \\ n\left(E\right)=16 \\ n\left(S\right)=52 \end{gathered}$$

Now substitute the values in the above formula we get

$$\begin{gathered} P\left(E\right)=\frac{16}{52} \\ P\left(E\right)=\frac{4}{13} \end{gathered}$$

The probability that a card selected at random from a standard deck of 52 cards is an ace or a heart is$P\left(E\right)=\frac{4}{13}$ .