One card is drawn at random from a standard pack of 52 playing cards. What is the probability that the card drawn is either a diamond or a king?

Introduction / Context:
This card probability question involves a union of two events based on suit and rank. We want the probability that a single card randomly drawn from a standard deck is either a diamond or a king. Because one specific card belongs to both groups, we must take overlap into account when counting favourable outcomes.

Given Data / Assumptions:

• A standard deck has 52 cards.
• There are 4 suits, so diamonds have 13 cards.
• There are 4 kings in the deck, one in each suit.
• The king of diamonds is both a king and a diamond.
• One card is drawn at random with all cards equally likely.
• Event A: card is a diamond. Event B: card is a king.

Concept / Approach:
To find the probability that the card is a diamond or a king, we use the union formula: P(A or B) = P(A) + P(B) minus P(A and B). In counting terms, that means we add the number of diamonds and the number of kings, then subtract one for the overlap, which is the king of diamonds. Finally, we divide by the total number of cards in the deck.

Step-by-Step Solution:
Number of diamonds = 13. Number of kings = 4. The king of diamonds is common to both sets. Favourable cards that are diamonds or kings = 13 + 4 - 1 = 16. Total cards in the deck = 52. Required probability = 16 / 52. Simplify 16 / 52 by dividing numerator and denominator by 4 to get 4 / 13.

Verification / Alternative check:
Compute probabilities directly. P(diamond) = 13 / 52. P(king) = 4 / 52. P(king of diamonds) = 1 / 52. So P(diamond or king) = 13 / 52 + 4 / 52 minus 1 / 52 = 16 / 52. This simplifies to 4 / 13, which matches the earlier combination based reasoning and confirms the answer.

Why Other Options Are Wrong:
5/12 would require about 21.67 favourable cards, which is not an integer count and does not match 16. 5/13 implies 20 favourable cards, again inconsistent with the count of 16. 3/14 is much smaller and would represent only approximately 11.14 favourable cards, which is not correct.

Common Pitfalls:
Learners often forget to subtract the overlapping card that belongs to both categories and simply add 13 and 4 to get 17 favourable cards. This overcounts the king of diamonds and leads to an incorrect probability. Always check whether an item can satisfy both conditions when applying the union formula in probability questions.

Final Answer:
The probability that the drawn card is a diamond or a king is 4/13.