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 a spade? (b) what is the probability that the card drawn is a six? (c) what is the probability that the card drawn is a spade or a six?

Explanation:

Step1: Recall probability formula

The probability of an event $E$ 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, $n(S) = 52$ (total number of cards in a deck).

Step2: Calculate probability of drawing a spade

There are 13 spades in a deck. So, for event $A$ (drawing a spade), $n(A)=13$. Then $P(A)=\frac{n(A)}{n(S)}=\frac{13}{52}=\frac{1}{4}$.

Step3: Calculate probability of drawing a six

There are 4 six - cards in a deck. So, for event $B$ (drawing a six), $n(B) = 4$. Then $P(B)=\frac{n(B)}{n(S)}=\frac{4}{52}=\frac{1}{13}$.

Step4: Calculate probability of drawing a spade or a six

The number of cards that are either a spade or a six: The number of spades is 13 and the number of six - cards is 4. But the six of spades is counted twice. So, $n(A\cup B)=n(A)+n(B)-n(A\cap B)$. Since $n(A\cap B) = 1$ (the six of spades), $n(A\cup B)=13 + 4-1=16$. Then $P(A\cup B)=\frac{n(A\cup B)}{n(S)}=\frac{16}{52}=\frac{4}{13}$.

Answer:

(a) $\frac{1}{4}$
(b) $\frac{1}{13}$
(c) $\frac{4}{13}$