To determine the probability of drawing a single card from a standard deck that is either a 3 or a diamond, follow these steps:
1. Identify the total number of cards in a deck:
A standard deck has 52 cards.
2. Count the number of 3s in the deck:
There are 4 suits (hearts, diamonds, clubs, spades) and each suit has one 3. So, there are 4 cards that are 3s in the deck.
3. Count the number of diamonds in the deck:
There are 13 diamonds in a deck (one for each rank from Ace to King).
4. Identify the overlap (cards that are both 3s and diamonds):
There is exactly one card that is both a 3 and a diamond, which is the 3 of diamonds.
5. Apply the principle of inclusion and exclusion:
To avoid double-counting the 3 of diamonds, we use the formula for the union of two sets:
[tex]\[
\text{Number of favorable outcomes} = (\text{Number of 3s}) + (\text{Number of diamonds}) - (\text{Number of 3s that are diamonds})
\][/tex]
Substituting the values:
[tex]\[
\text{Number of favorable outcomes} = 4 + 13 - 1 = 16
\][/tex]
6. Calculate the probability:
The probability is the number of favorable outcomes divided by the total number of possible outcomes (total cards):
[tex]\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of cards}} = \frac{16}{52}
\][/tex]
7. Simplify the fraction:
The fraction [tex]\(\frac{16}{52}\)[/tex] simplifies to [tex]\(\frac{4}{13}\)[/tex] when divided by their greatest common divisor, which is 4.
Therefore, the probability of drawing a single card from a standard deck of 52 cards that is either a 3 or a diamond is [tex]\(\frac{4}{13}\)[/tex].
So, the correct answer is:
[tex]\(\frac{4}{13}\)[/tex]
