A single card is drawn at random from a standard pack of 52 playing cards. What is the probability that the card drawn is either black or a king (or both)?

Introduction / Context:
This is a probability question involving a standard deck of playing cards and a union of two events: drawing a black card or drawing a king. Such problems are a good test of understanding of the addition rule for probabilities, especially the need to subtract the overlap when events are not mutually exclusive. The classical deck has 52 cards divided evenly by colour and suit, and kings are a well-known subset in each suit.

Given Data / Assumptions:
- Standard deck: 52 cards in total.

- There are 26 black cards (13 spades and 13 clubs).

- There are 4 kings in total (one in each suit: hearts, diamonds, clubs, spades).

- Among these 4 kings, 2 are black (king of spades and king of clubs) and 2 are red (king of hearts and king of diamonds).

- We want P(black or king), where "or" is inclusive: black, king, or both.

Concept / Approach:
We use the addition rule for probabilities of two events A and B:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

Here, let A be the event "card is black", and B be the event "card is a king". We first count how many cards satisfy each individual event, then count how many satisfy both (black kings), and finally substitute into the formula to avoid double-counting the intersection.

Step-by-Step Solution:
Step 1: Count black cards: there are 26 black cards out of 52.Step 2: So P(A) = 26/52.Step 3: Count kings: there are 4 kings out of 52 cards.Step 4: So P(B) = 4/52.Step 5: Count black kings: there are 2 black kings (clubs and spades).Step 6: So P(A ∩ B) = 2/52.Step 7: Apply the addition rule: P(A ∪ B) = 26/52 + 4/52 - 2/52.Step 8: Combine the fractions: (26 + 4 - 2) / 52 = 28/52.Step 9: Simplify 28/52 by dividing numerator and denominator by 4: 28/52 = 7/13.

Verification / Alternative check:
We can also count favourable cards directly. A card is favourable if it is black (26 cards) and not a king (26 - 2 = 24 cards), or is a king of any colour (4 cards). So total favourable = 24 non-king black cards + 4 kings = 28 cards. Dividing by 52 gives 28/52 = 7/13, which matches our earlier calculation using the addition rule. This double check confirms both counting methods agree.

Why Other Options Are Wrong:
- 15/26 corresponds to (26 + 4) / 52 without subtracting the overlap of 2 black kings, so it double-counts those cards.

- 15/52 and 17/26 do not arise from any correct application of the counting or probability formulas for this problem; they reflect miscounting or mis-simplified fractions.

Common Pitfalls:
A common mistake is forgetting to subtract P(A ∩ B) when events overlap, leading to double-counting the black kings. Another error is to misinterpret "either black or a king" as exclusive or, which would require subtracting the intersection instead of adding and then subtracting. Here, standard probability language uses inclusive or, so we must include black kings exactly once in the count.

Final Answer:
The probability that the card drawn is either black or a king is 7/13.