To determine the probability that a card drawn from a standard deck of 52 playing cards will be either a spade or a seven, let's break down the problem step by step:
1. Identify the total number of cards in the deck:
- A standard deck contains 52 cards.
2. Count the number of spades in the deck:
- There are 13 spades in a deck.
3. Count the number of sevens in the deck:
- There are 4 sevens in a deck.
4. Identify the overlap between spades and sevens:
- There is precisely one card that is both a spade and a seven, which is the seven of spades.
5. Calculate the individual probabilities:
- Probability of drawing a spade (P(A)):
[tex]\[
P(\text{spade}) = \frac{\text{number of spades}}{\text{total number of cards}} = \frac{13}{52} = 0.25
\][/tex]
- Probability of drawing a seven (P(B)):
[tex]\[
P(\text{seven}) = \frac{\text{number of sevens}}{\text{total number of cards}} = \frac{4}{52} \approx 0.0769
\][/tex]
- Probability of drawing the seven of spades (P(A and B)):
[tex]\[
P(\text{spade and seven}) = \frac{\text{number of seven of spades}}{\text{total number of cards}} = \frac{1}{52} \approx 0.0192
\][/tex]
6. Apply the formula for the probability of drawing a spade or a seven (P(A or B)):
[tex]\[
P(\text{spade or seven}) = P(\text{spade}) + P(\text{seven}) - P(\text{spade and seven})
\][/tex]
Substituting the values:
[tex]\[
P(\text{spade or seven}) = 0.25 + 0.0769 - 0.0192 \approx 0.3077
\][/tex]
Therefore, the probability that a card drawn from a standard deck will be either a spade or a seven is approximately [tex]\(0.3077\)[/tex].
