A standard deck contains 52 cards with 4 suits (spades, hearts, diamonds, and clubs) with 13 different denominations (A, K, Q, J, 10, 9, …, 2).
What is the probability that a single card drawn randomly from the deck is a “King” ?
What is the probability that a single card drawn randomly from the deck is not a “King” ?
What is the probability that a single card drawn randomly from the deck is a “Spade” ?
What is the probability that a single card drawn randomly from the deck is not a “Spade” ?
What is the probability that a single card drawn randomly from the deck is a “King” or a “Spade” ?
Use the Addition Rule of Probability: P(A or B)=P(A)+P(B)-P(A and B).
Are the events “King” and “Spade” mutually exclusive? Explain why or why not.
The probability that a single card drawn randomly from the deck is a "King" is 4/52 or 1/13. This is because there are 4 Kings in a deck of 52 cards, so the probability is the number of favorable outcomes (4) divided by the total number of possible outcomes (52).
The probability that a single card drawn randomly from the deck is not a "King" is 1 - (4/52) or 48/52. This is because there are 48 cards that are not Kings in a deck of 52 cards, so the probability is 1 minus the probability of drawing a King.
The probability that a single card drawn randomly from the deck is a "Spade" is 13/52 or 1/4. This is because there are 13 Spades in a deck of 52 cards, so the probability is the number of favorable outcomes (13) divided by the total number of possible outcomes (52).
The probability that a single card drawn randomly from the deck is not a "Spade" is 1 - (13/52) or 39/52. This is because there are 39 cards that are not Spades in a deck of 52 cards, so the probability is 1 minus the probability of drawing a Spade.
The probability that a single card drawn randomly from the deck is a "King" or a "Spade" can be calculated using the Addition Rule of Probability:
P(King or Spade) = P(King) + P(Spade) - P(King and Spade)
= (4/52) + (13/52) - (1/52)
= 16/52
= 4/13
The events "King" and "Spade" are not mutually exclusive. This is because there is one card in the deck that is both a King and a Spade (the King of Spades). If the events were mutually exclusive, it would mean that there are no cards that satisfy both conditions.
To find the probability for each question, we can use the formula:
Probability = Number of favorable outcomes / Total number of possible outcomes
1. Probability of drawing a King:
There are 4 Kings in the deck, one for each suit. Therefore, there are 4 favorable outcomes (the 4 Kings) out of a total of 52 possible outcomes.
So, the probability of drawing a King is 4/52, which simplifies to 1/13.
2. Probability of not drawing a King:
Since there are 4 Kings in the deck, there are 48 cards that are not Kings. So, there are 48 favorable outcomes out of a total of 52 possible outcomes.
The probability of not drawing a King is 48/52, which simplifies to 12/13.
3. Probability of drawing a Spade:
There are 13 Spades in the deck, one for each denomination. Therefore, there are 13 favorable outcomes (the 13 Spades) out of a total of 52 possible outcomes.
So, the probability of drawing a Spade is 13/52, which simplifies to 1/4.
4. Probability of not drawing a Spade:
Since there are 13 Spades in the deck, there are 39 cards that are not Spades. So, there are 39 favorable outcomes out of a total of 52 possible outcomes.
The probability of not drawing a Spade is 39/52, which simplifies to 3/4.
5. Probability of drawing a King or a Spade:
To calculate this probability, we can use the Addition Rule of Probability:
P(King or Spade) = P(King) + P(Spade) - P(King and Spade)
From the previous calculations, P(King) = 1/13 and P(Spade) = 1/4. To find P(King and Spade), we know that there is only one card in the deck that is both a King and a Spade, which is the King of Spades.
Therefore, P(King and Spade) = 1/52.
Substituting the values into the formula:
P(King or Spade) = 1/13 + 1/4 - 1/52
= 4/52 + 13/52 - 1/52
= 16/52
= 4/13
So, the probability of drawing a King or a Spade is 4/13.
6. "King" and "Spade" as mutually exclusive events:
No, the events "King" and "Spade" are not mutually exclusive. This is because there is one card in the deck that is both a King and a Spade, which is the King of Spades. Mutually exclusive events cannot occur at the same time, but in this case, it is possible to draw a card that is both a King and a Spade.
To find the probability of each event, we need to first understand the total number of possible outcomes and the number of favorable outcomes.
1. Probability of drawing a "King":
In a standard deck, there are 4 Kings (one for each suit). So, the total number of favorable outcomes is 4. Since there are 52 cards in total, the probability of drawing a King is 4/52, which simplifies to 1/13.
2. Probability of not drawing a "King":
Since there are 4 Kings in the deck, there are 52 - 4 = 48 non-King cards. So, the total number of favorable outcomes is 48. The probability of not drawing a King is 48/52, which simplifies to 12/13.
3. Probability of drawing a "Spade":
In a standard deck, there are 13 Spades (one for each denomination). So, the total number of favorable outcomes is 13. The probability of drawing a Spade is 13/52, which simplifies to 1/4.
4. Probability of not drawing a "Spade":
Since there are 13 Spades in the deck, there are 52 - 13 = 39 non-Spade cards. So, the total number of favorable outcomes is 39. The probability of not drawing a Spade is 39/52, which simplifies to 3/4.
5. Probability of drawing a "King" or a "Spade":
To find this probability, we use the Addition Rule of Probability:
P(King or Spade) = P(King) + P(Spade) - P(King and Spade).
P(King) = 1/13 (as calculated earlier)
P(Spade) = 1/4 (as calculated earlier)
For a card to be both a King and a Spade, it must be the King of Spades, of which there is only 1 in the deck. So, P(King and Spade) = 1/52.
Plugging these values into the formula, we get:
P(King or Spade) = 1/13 + 1/4 - 1/52 = 4/13 + 13/52 - 1/52 = 16/52 = 4/13.
Finally, regarding the events "King" and "Spade":
Two events are considered mutually exclusive if they cannot occur at the same time. In this case, a card cannot be both a King and a Spade. Since there is only one King of Spades in the deck, the events "King" and "Spade" are mutually exclusive.