From a standard pack of 52 playing cards, one card is drawn at random. What is the probability that the card drawn is either a spade or an ace?

Introduction / Context:
This question tests your understanding of probability with overlapping events. You are asked to find the probability of drawing a card that is either a spade or an ace from a full deck. Since one card, the ace of spades, belongs to both categories, you must correctly apply the addition rule for probabilities with overlap.

Given Data / Assumptions:

• Total number of cards in the deck = 52.
• Number of spade cards = 13.
• Number of aces in the deck = 4.
• The ace of spades is counted in both the spade set and the ace set.
• One card is drawn at random with all cards equally likely.

Concept / Approach:
For two events A and B, the probability that at least one occurs is: P(A or B) = P(A) + P(B) - P(A and B). Here:

• A = card is a spade.
• B = card is an ace.
• A and B = card is the ace of spades.

We will first count how many cards are in each category and then convert those counts into probabilities.

Step-by-Step Solution:
Step 1: Number of spades in the deck = 13. Step 2: Number of aces in the deck = 4. Step 3: The ace of spades is common to both sets, so the intersection has size 1. Step 4: Convert counts to probabilities. P(spade) = 13 / 52. P(ace) = 4 / 52. P(ace of spades) = 1 / 52. Step 5: Apply the addition rule for overlapping events. P(spade or ace) = 13/52 + 4/52 - 1/52. Step 6: Simplify the numerator: (13 + 4 - 1) / 52 = 16 / 52. Step 7: Simplify 16 / 52 by dividing numerator and denominator by 4 to get 4 / 13.

Verification / Alternative check:
You can verify by direct counting. Count all spades (13 cards) and all non spade aces. The non spade aces are hearts, diamonds, and clubs, which are 3 additional cards. So the total favorable cards = 13 + 3 = 16. Total cards = 52. Probability = 16 / 52 = 4 / 13, which matches the previous calculation using the formula with intersection.

Why Other Options Are Wrong:
Option 5/13 is larger than the correct value and could arise from adding 13/52 and 4/52 without subtracting the overlap, which double counts the ace of spades. Option 1/4 would be correct if there were 13 favorable cards instead of 16. Option 3/13 underestimates the probability and does not correspond to any correct tally of favorable cards. Option 7/26 also does not simplify to the correct fraction and does not align with the actual count of 16 favorable cards.

Common Pitfalls:
A very common mistake is to ignore intersections when using the addition rule. Whenever two conditions can occur together in at least one outcome, you must subtract the intersection once. Another error is forgetting the exact composition of a standard deck or miscounting the number of aces. Always double check how many cards satisfy each condition and whether there is any overlap between those sets before you convert the counts into probabilities.

Final Answer:
The probability that the card drawn is either a spade or an ace is 4/13.