Question

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 heart or 7? (your answer must be in the form of a reduced fraction.) question help: video 1 video 2 message instructor submit question

Explanation:

Step1: Recall Probability Formula

For two events \( A \) and \( B \), the probability of \( A \) or \( B \) is \( P(A \cup B)=P(A)+P(B)-P(A \cap B) \). Let \( A \) be the event of selecting a heart, and \( B \) be the event of selecting a 7.

Step2: Calculate \( P(A) \)

A standard deck has 52 cards, and there are 13 hearts. So \( P(A)=\frac{13}{52} \).

Step3: Calculate \( P(B) \)

There are 4 sevens (one in each suit). So \( P(B)=\frac{4}{52} \).

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

The card that is both a heart and a 7 is the 7 of hearts, so there's 1 such card. Thus, \( P(A \cap B)=\frac{1}{52} \).

Step5: Compute \( P(A \cup B) \)

Substitute into the formula: \( P(A \cup B)=\frac{13}{52}+\frac{4}{52}-\frac{1}{52}=\frac{13 + 4-1}{52}=\frac{16}{52} \). Reduce the fraction: \( \frac{16\div4}{52\div4}=\frac{4}{13} \).

Answer:

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