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 face card? (b) what is the probability that the card drawn is a diamond? (c) what is the probability that the card drawn is a face card or a diamond?

Explanation:

Step1: Calculate the probability of a face card

There are 12 face cards (4 suits × 3 face cards per suit) in a 52 - card deck.
The probability formula is \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
So, \(P(\text{face card})=\frac{12}{52}=\frac{3}{13}\)

Step2: Calculate the probability of a diamond

There are 13 diamonds in a 52 - card deck.
Using the probability formula \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\), we get \(P(\text{diamond})=\frac{13}{52}=\frac{1}{4}\)

Step3: Calculate the probability of a face card or a diamond

Use the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
\(P(\text{face card})=\frac{12}{52}\), \(P(\text{diamond})=\frac{13}{52}\)
The number of cards that are both face cards and diamonds is 3 ( \(J\diamondsuit\), \(Q\diamondsuit\), \(K\diamondsuit\) )
\(P(\text{face card}\cap\text{diamond})=\frac{3}{52}\)
\(P(\text{face card}\cup\text{diamond})=\frac{12 + 13- 3}{52}=\frac{22}{52}=\frac{11}{26}\)

Answer:

(a) \(\frac{3}{13}\)
(b) \(\frac{1}{4}\)
(c) \(\frac{11}{26}\)