Question

a standard deck of cards contains 52 cards. thirteen of the cards are spades, and one of the spades is a king. in all, the deck contains 4 kings. when drawing one card at random from a shuffled deck, let a = {getting a king} and b = {getting a spade}. what is the probability a or b happens?
○ 1/52
○ 4/52 + 13/52 - 1/52
○ 4/52 + 13/52
○ 4/52

Explanation:

Step1: Recall the formula for the probability of the union of two events

The formula for \( P(A \cup B) \) (the probability that \( A \) or \( B \) occurs) is \( P(A) + P(B) - P(A \cap B) \), where \( P(A \cap B) \) is the probability that both \( A \) and \( B \) occur.

Step2: Calculate \( P(A) \)

Event \( A \) is "getting a king". There are 4 kings in a deck of 52 cards, so \( P(A) = \frac{4}{52} \).

Step3: Calculate \( P(B) \)

Event \( B \) is "getting a spade". There are 13 spades in a deck of 52 cards, so \( P(B) = \frac{13}{52} \).

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

The event \( A \cap B \) is "getting a king of spades". There is 1 king of spades in the deck, so \( P(A \cap B) = \frac{1}{52} \).

Step5: Apply the union formula

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

Answer:

The option \( \boldsymbol{\frac{4}{52} + \frac{13}{52} - \frac{1}{52}} \) (the third option from the left)