Question

here is a table showing all 52 cards in a standard deck. suppose one card is drawn at random from a standard deck. answer each part. write your answers as fractions. (a) what is the probability that the card drawn is an ace? (b) what is the probability that the card drawn is a red card? (c) what is the probability that the card drawn is an ace or a red card?

Explanation:

Step1: Recall probability formula

The probability formula is $P(E)=\frac{n(E)}{n(S)}$, where $n(E)$ is the number of elements in the event - set and $n(S)$ is the number of elements in the sample - set. Here, $n(S) = 52$ (the number of cards in a standard deck).

Step2: Calculate probability of drawing an ace

There are 4 aces in a standard deck. So, $n(E_{ace})=4$. Then $P(\text{ace})=\frac{n(E_{ace})}{n(S)}=\frac{4}{52}=\frac{1}{13}$.

Step3: Calculate probability of drawing a red card

There are 26 red cards in a standard deck ($13$ hearts and $13$ diamonds). So, $n(E_{red}) = 26$. Then $P(\text{red})=\frac{n(E_{red})}{n(S)}=\frac{26}{52}=\frac{1}{2}$.

Step4: Calculate probability of drawing an ace or a red card

Use the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. The number of red - aces is 2 (ace of hearts and ace of diamonds). So, $P(A\cap B)=\frac{2}{52}$.
$P(\text{ace}\cup\text{red})=P(\text{ace})+P(\text{red})-P(\text{ace}\cap\text{red})=\frac{4}{52}+\frac{26}{52}-\frac{2}{52}=\frac{4 + 26-2}{52}=\frac{28}{52}=\frac{7}{13}$.

Answer:

(a) $\frac{1}{13}$
(b) $\frac{1}{2}$
(c) $\frac{7}{13}$