A single card is drawn at random from a well shuffled standard deck of 52 cards. What is the probability of getting either the queen of clubs or the king of hearts?

Introduction / Context:
This question is a simple probability problem involving a standard deck of cards. We are asked to find the probability that a card drawn at random is either the queen of clubs or the king of hearts.

Given Data / Assumptions:

• A standard deck has 52 cards.
• There is exactly one queen of clubs in the deck.
• There is exactly one king of hearts in the deck.
• The deck is well shuffled, so every card is equally likely to be drawn.
• The two desired cards are distinct.

Concept / Approach:
We are dealing with a union of two mutually exclusive events: drawing the queen of clubs or drawing the king of hearts. Since a single card cannot be both at the same time, the probability of their union is the sum of their individual probabilities. Each specific card has probability 1/52 of being drawn.

Step-by-Step Solution:
Probability of drawing the queen of clubs = 1 / 52.Probability of drawing the king of hearts = 1 / 52.Because these two cards are different, the events are mutually exclusive when drawing a single card.Therefore, probability of drawing either the queen of clubs or the king of hearts = 1/52 + 1/52 = 2/52.Simplify 2/52 by dividing numerator and denominator by 2 to get 1/26.

Verification / Alternative check:
We can think of the sample space as 52 equally likely cards and the favourable outcomes as a set of 2 specific cards. The probability is simply the size of the favourable set divided by the size of the sample space, which is 2 / 52 = 1 / 26. This aligns with the previous calculation using the addition rule for mutually exclusive events.

Why Other Options Are Wrong:
The value 1/13 would correspond to 4 favourable cards out of 52, which would be the case if we were looking for all queens of a suit or all kings of a suit, but we are concerned with exactly two specific cards. The fraction 2/13 arises from miscounting favourable cards or misapplying simplification. The probability 1/52 would correspond to only one specific card being favourable. Only 1/26 correctly reflects two favourable cards out of 52.

Common Pitfalls:
Students sometimes incorrectly think there are four queens of clubs or four kings of hearts, confusing suits and ranks. Another error is to forget to simplify 2/52 to 1/26, although the unsimplified fraction is still numerically correct. Keeping track of the deck structure (4 suits and 13 ranks) is helpful for avoiding such mistakes.

Final Answer:
The probability of drawing the queen of clubs or the king of hearts is 1/26.