In a standard deck of cards, what is the probability of drawing an ace OR a black card (answer choices are in the form of a percentage, rounded to the nearest whole number)?
A.) 25%
B.) 50%
C.) 54%
D.) 33%
The correct answer and explanation is:
Correct Answer: C.) 54%
To determine the probability of drawing an ace or a black card from a standard deck of 52 cards, we use the principle of inclusion and exclusion. This principle avoids counting overlapping outcomes more than once.
There are two parts to this problem:
- Number of aces in a deck:
There are 4 aces in a standard deck (one per suit). - Number of black cards in a deck:
There are 26 black cards (13 spades and 13 clubs).
However, 2 of the aces are black (Ace of spades and Ace of clubs), so if we simply added the number of aces and black cards (4 + 26), we would be double-counting those 2 black aces.
So we apply the formula:
P(A or B) = P(A) + P(B) – P(A and B)
Let:
- A = drawing an ace = 4 cards
- B = drawing a black card = 26 cards
- A and B = drawing a black ace = 2 cards
So,
- Favorable outcomes = 4 (aces) + 26 (black cards) – 2 (black aces) = 28
Now we calculate the probability:
P(ace or black card) = 28 / 52
Convert to a percentage:
28 ÷ 52 ≈ 0.5385 → 53.85%, which rounds to 54%
Therefore, the correct answer is C.) 54%
This means that in a single random draw from a standard deck, there is a 54 percent chance that the card drawn will either be an ace or a black card. Understanding how overlapping categories work is essential in probability, especially when dealing with sets that are not mutually exclusive.