To solve the problem of finding the probability that a card drawn from a standard deck of 52 playing cards will be a spade or a seven, we can use the addition rule for probability.
The addition rule for probability states:
[tex]\[
P(A \text { or } B) = P(A) + P(B) - P(A \text { and } B)
\][/tex]
Here's a step-by-step solution:
1. Identify the Total Number of Cards: There are 52 cards in a standard deck.
2. Number of Spades in the Deck: There are 13 spade cards in a deck (one for each rank 2 through Ace).
3. Number of Sevens in the Deck: There are 4 seven cards in a deck (one from each suit: hearts, diamonds, clubs, and spades).
4. Number of Seven of Spades: Since seven of spades is counted in both the spades and the sevens, we need to account for it only once. There is exactly 1 seven of spades.
5. Calculate the Probabilities:
- Probability of Drawing a Spade [tex]\(P(A)\)[/tex]:
[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 [tex]\(P(B)\)[/tex]:
[tex]\[
P(\text{Seven}) = \frac{\text{Number of Sevens}}{\text{Total Number of Cards}} = \frac{4}{52} \approx 0.0769
\][/tex]
- Probability of Drawing a Seven of Spades [tex]\(P(A \text { and } B)\)[/tex]:
[tex]\[
P(\text{Seven and Spade}) = \frac{\text{Number of Seven of Spades}}{\text{Total Number of Cards}} = \frac{1}{52} \approx 0.0192
\][/tex]
6. Apply the Addition Rule for Probability:
[tex]\[
P(\text {Spade or Seven}) = P(\text {Spade}) + P(\text {Seven}) - P(\text {Seven and Spade})
\][/tex]
Substituting the calculated probabilities:
[tex]\[
P(\text {Spade or Seven}) = 0.25 + 0.0769 - 0.0192
\][/tex]
7. Sum Up the Probabilities:
[tex]\[
P(\text {Spade or Seven}) = 0.3077
\][/tex]
Therefore, the probability that a card drawn from a standard deck will be either a spade or a seven, expressed as a decimal rounded to four decimal places, is:
[tex]\[
\boxed{0.3077}
\][/tex]
