Question

question 10
if you randomly select a card from a well - shuffled standard deck of 52 cards, what is the probability that
the card you select is a spade or 8? (your answer must be in the form of a reduced fraction.)

Explanation:

Step1: Recall Probability Formula

The probability of an event \( A \) or \( B \) is given by the formula \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), where \( A \) is the event of selecting a spade and \( B \) is the event of selecting an 8.

Step2: Calculate \( P(A) \)

A standard deck has 52 cards. There are 13 spades (since there are 4 suits and 13 cards per suit). So, \( P(A)=\frac{13}{52} \).

Step3: Calculate \( P(B) \)

There are 4 eights in a deck (one for each suit). So, \( P(B)=\frac{4}{52} \).

Step4: Calculate \( P(A \cap B) \)

The intersection of spades and eights is the 8 of spades, so there is 1 such card. Thus, \( P(A \cap B)=\frac{1}{52} \).

Step5: Apply the Formula

\( P(A \cup B)=\frac{13}{52}+\frac{4}{52}-\frac{1}{52}=\frac{13 + 4 - 1}{52}=\frac{16}{52} \). Simplify the fraction by dividing numerator and denominator by 4: \( \frac{16\div4}{52\div4}=\frac{4}{13} \).

Answer:

\(\frac{4}{13}\)