One card is drawn at random from a standard pack of 52 playing cards. What is the probability that the card drawn is neither a spade nor a king?

Introduction / Context:
This question tests your ability to use complements and the inclusion exclusion principle with a standard deck of cards. You must find the probability that a randomly drawn card is neither a spade nor a king. Rather than counting directly, it is much easier to find the probability of the card being a spade or a king and then subtract that from 1.

Given Data / Assumptions:

• A standard deck has 52 cards.
• There are 4 suits: spades, hearts, diamonds, clubs.
• Each suit has 13 cards.
• There are 4 kings in the deck, one in each suit.
• We need the probability that a drawn card is not a spade and not a king.

Concept / Approach:
Let A be the event that the card is a spade and B be the event that the card is a king. We first find P(A or B), the probability that the card is either a spade or a king (or both). Then we use: P(neither A nor B) = 1 - P(A or B). To get P(A or B), we apply the formula: P(A or B) = P(A) + P(B) - P(A and B).

Step-by-Step Solution:
Step 1: Number of spade cards = 13, so P(A) = 13 / 52. Step 2: Number of kings in the deck = 4, so P(B) = 4 / 52. Step 3: One card, the king of spades, is both a spade and a king, so P(A and B) = 1 / 52. Step 4: Compute P(A or B) using inclusion exclusion. P(A or B) = 13/52 + 4/52 - 1/52 = 16/52. Step 5: Simplify 16/52 by dividing numerator and denominator by 4 to get 4/13. Step 6: Now compute P(neither A nor B). P(neither spade nor king) = 1 - 4/13. Step 7: Convert 1 to 13/13 and subtract: 13/13 - 4/13 = 9/13.

Verification / Alternative check:
You can also verify by direct counting. First, count how many cards you must exclude: all spades (13 cards) and any remaining kings that are not spades (3 more cards). That is 13 + 3 = 16 cards that are either spades or non spade kings. The remaining cards in the deck are 52 - 16 = 36. Therefore, there are 36 cards that are neither spades nor kings. The probability is 36 / 52, which simplifies to 9 / 13. This matches the complement based calculation.

Why Other Options Are Wrong:
Option 4/13 is the probability of drawing a spade or a king, not the probability of avoiding both. Option 1/2 suggests that exactly half the deck is excluded, which is not accurate. Option 3/13 and option 1/13 are too small and imply that almost all cards are spades or kings, which is clearly not the case. None of these match the correct count of 36 safe cards out of 52 total.

Common Pitfalls:
One frequent mistake is to add P(spade) and P(king) and forget to subtract P(king of spades), thereby double counting that card. Another mistake is to attempt to directly count the cards that are neither spade nor king without systematically subtracting the excluded cards. Using complements and inclusion exclusion reduces the chance of such errors and is a powerful method for many exam style probability problems.

Final Answer:
The probability that the card drawn is neither a spade nor a king is 9/13.