Question

question 6
a card is drawn randomly from a standard 52 - card deck. find the probability of the given event.
write your answers as simplified fractions.
a. the card drawn is 8.
p(8) is equal to
b. the card drawn is the 8 of clubs.
p(8 of clubs) is equal to
c. the card drawn is a face card (jack, queen, or king).
p(face card) is equal to
d. the card drawn is not a diamond.
p(not a diamond) is equal to
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Explanation:

Step1: Calculate probability for part A

Probability formula is \(P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
Total number of cards \(n = 52\).
Number of 8 - cards: There are 4 suits (hearts, diamonds, clubs, spades), so number of favorable outcomes \(m_1=4\).
\(P(8)=\frac{4}{52}=\frac{1}{13}\)

Step2: Calculate probability for part B

Number of 8 of clubs: There is only 1 card which is 8 of clubs.
Using the probability formula \(P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\), with \(n = 52\) and number of favorable outcomes \(m_2 = 1\).
\(P(8\text{ of clubs})=\frac{1}{52}\)

Step3: Calculate probability for part C

Number of face - cards: Jack, Queen, King for each of the 4 suits. So number of face - cards \(m_3=3\times4 = 12\).
Using the probability formula \(P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\) with \(n = 52\).
\(P(\text{face card})=\frac{12}{52}=\frac{3}{13}\)

Step4: Calculate probability for part D

Number of diamond cards: There are 13 diamond cards. Number of non - diamond cards \(m_4=52 - 13=39\).
Using the probability formula \(P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\) with \(n = 52\).
\(P(\text{not a diamond})=\frac{39}{52}=\frac{3}{4}\)

Answer:

A. \(\frac{1}{13}\)
B. \(\frac{1}{52}\)
C. \(\frac{3}{13}\)
D. \(\frac{3}{4}\)