Answer :
Let's explore the probabilities step by step.
### Part (a): Probability of drawing a spade or a king
First, remember that there are 52 cards in a standard deck.
1. Number of spades: There are 13 spades in a standard deck.
2. Number of kings: There are 4 kings in a deck.
3. Number of spade kings: Since the king of spades is counted in both the spade set and king set, we must avoid counting it twice.
Therefore, the number of favorable outcomes for drawing a spade or a king:
[tex]\[ 13 \ (\text{spades}) + 4 \ (\text{kings}) - 1 \ (\text{king of spades}) = 16 \][/tex]
The probability:
[tex]\[ \frac{16}{52} = \frac{4}{13} \approx 0.3077 \][/tex]
### Part (b): Probability of drawing neither a jack nor a king
1. Number of jacks: 4 jacks in a deck.
2. Number of kings: 4 kings in a deck.
There are 8 cards that are either a jack or a king. Consequently, the number of cards that are neither jack nor king:
[tex]\[ 52 - 8 = 44 \][/tex]
The probability:
[tex]\[ \frac{44}{52} = \frac{11}{13} \approx 0.8462 \][/tex]
### Part (c): Probability of drawing a club and queen
1. Number of queen of clubs: There is exactly 1 queen of clubs.
The probability:
[tex]\[ \frac{1}{52} \approx 0.0192 \][/tex]
In summary, the probabilities are:
- Drawing a spade or a king: [tex]\( \approx 0.3077 \)[/tex]
- Drawing neither a jack nor a king: [tex]\( \approx 0.8462 \)[/tex]
- Drawing a club and queen: [tex]\( \approx 0.0192 \)[/tex]
### Part (a): Probability of drawing a spade or a king
First, remember that there are 52 cards in a standard deck.
1. Number of spades: There are 13 spades in a standard deck.
2. Number of kings: There are 4 kings in a deck.
3. Number of spade kings: Since the king of spades is counted in both the spade set and king set, we must avoid counting it twice.
Therefore, the number of favorable outcomes for drawing a spade or a king:
[tex]\[ 13 \ (\text{spades}) + 4 \ (\text{kings}) - 1 \ (\text{king of spades}) = 16 \][/tex]
The probability:
[tex]\[ \frac{16}{52} = \frac{4}{13} \approx 0.3077 \][/tex]
### Part (b): Probability of drawing neither a jack nor a king
1. Number of jacks: 4 jacks in a deck.
2. Number of kings: 4 kings in a deck.
There are 8 cards that are either a jack or a king. Consequently, the number of cards that are neither jack nor king:
[tex]\[ 52 - 8 = 44 \][/tex]
The probability:
[tex]\[ \frac{44}{52} = \frac{11}{13} \approx 0.8462 \][/tex]
### Part (c): Probability of drawing a club and queen
1. Number of queen of clubs: There is exactly 1 queen of clubs.
The probability:
[tex]\[ \frac{1}{52} \approx 0.0192 \][/tex]
In summary, the probabilities are:
- Drawing a spade or a king: [tex]\( \approx 0.3077 \)[/tex]
- Drawing neither a jack nor a king: [tex]\( \approx 0.8462 \)[/tex]
- Drawing a club and queen: [tex]\( \approx 0.0192 \)[/tex]
