Question

A single card is drawn at random from a shuffled deck. What is the probability that it is red? That it is the ace of hearts? That it is either a three or a five? That it is either an ace or red or both?

Short answer

The probability that it is red is $\frac{1}{2}$.

The probability that it is ace of heart is$\frac{1}{52}$.

The probability that it is three or five is $\frac{2}{13}$.

The probability that it is either an ace or red or both is$\frac{7}{13}$

Step-by-step solution

Definition of Probability

The probability of any event is defined as the ratio of the number of outcomes associated with the event to the total number of possible outcomes.

The formula for probability of any event E is expressed as follows,

$\mathit{P}\mathbf{=}\frac{\mathbf{n}\mathbf{u}\mathbf{m}\mathbf{b}\mathbf{e}\mathbf{r}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{o}\mathbf{u}\mathbf{t}\mathbf{c}\mathbf{o}\mathbf{m}\mathbf{e}\mathbf{s}\mathbf{}\mathbf{f}\mathbf{a}\mathbf{v}\mathbf{o}\mathbf{r}\mathbf{a}\mathbf{b}\mathbf{l}\mathbf{e}\mathbf{}\mathbf{t}\mathbf{o}\mathbf{}\mathbf{E}}{\mathbf{t}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}\mathbf{}\mathbf{n}\mathbf{u}\mathbf{m}\mathbf{b}\mathbf{e}\mathbf{r}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{o}\mathbf{u}\mathbf{t}\mathbf{c}\mathbf{o}\mathbf{m}\mathbf{e}\mathbf{s}}$ ....(i)

Determination of the probability that the card drawn is red

In a deck of card, there are 52 cards of 4 suits namely spade, club, diamond and heart, out of which spade and club are black and diamond and heart are red.

There are 13 cards of each suit and 26 cards of each color.

This implies that the number of outcomes favourable are 26 and total number of outcomes are 52.

Find the probability that the card drawn is red by substituting the values in equation (i).

$$\begin{gathered} P=\frac{26}{52} \\ =\frac{1}{2} \end{gathered}$$

Determination of the probability that the card drawn is ace of heart

In a deck of card, there are 4 aces out of which only one is ace of heart.

This implies that the number of outcomes favourable are 1 and total number of outcomes are 52.

Find the probability that the card drawn is ace of heart by substituting the values in equation (i).

$P=\frac{1}{52}$

Determination of the probability that the card drawn is 3 or 5

In a deck of card, there are 4 threes and 4 fives.

This implies that the number of outcomes favorable are 8 and total number of outcomes are 52.

Find the probability that the card drawn is 3 or 5 by substituting the values in equation (i).

$P=\frac{8}{52}=\frac{2}{13}$

Determination of the probability that the card drawn is either an ace or red or both

In a deck of card, there are 4 aces and 26 red cards out of which 2 aces are red, this implies that total number of possibilities for a card to be an ace or red or both is .

This implies that the number of outcomes favorable are 28 and total number of outcomes are 52.

Find the probability that the card drawn is either an ace or red or both by substituting the values in equation (i).

$P=\frac{28}{52}=\frac{7}{13}$

Thus, the probability that it is red is $\frac{1}{2}$,the probability that it is ace of heart is $\frac{1}{52}$, the probability that it is three or five is $\frac{2}{13}$and the probability that it is either an ace or red or both is $\frac{7}{13}$.