A card is drawn from a well-shuffled deck of 52 cards. What is the probability that it is queen of spade?
Understanding Probability with a Deck of Cards
The question asks for the probability of drawing a specific card, the queen of spade, from a standard well-shuffled deck of 52 cards. Probability is a measure of the likelihood of an event occurring. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.
Total Possible Outcomes
A standard deck of cards has 52 cards. When we draw a single card from the deck, there are 52 different cards that could possibly be drawn. Therefore, the total number of possible outcomes is 52.
Favorable Outcomes: Drawing the Queen of Spade
We are interested in the specific event of drawing the queen of spade. In a standard deck of cards, there is only one queen of spade. So, the number of favorable outcomes for this event is 1.
A standard deck of cards contains the following cards:
- 4 Suits: Hearts, Diamonds, Clubs, Spades.
- Each suit has 13 cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King.
Thus, there is one Queen in the Spades suit, which is the Queen of Spade.
Calculating the Probability
The formula for probability is:
\( P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)
In this case, the event is drawing the queen of spade.
- Number of favorable outcomes (drawing the queen of spade) = 1
- Total number of possible outcomes (total cards in the deck) = 52
So, the probability of drawing the queen of spade is:
\( P(\text{Queen of Spade}) = \frac{1}{52} \)
Comparing with Options
Let's compare our calculated probability with the given options:
- Option 1: \(\frac{1}{{52}}\)
- Option 2: \(\frac{1}{{13}}\) (This would be the probability of drawing any Queen or any Spade)
- Option 3: \(\frac{1}{4}\) (This would be the probability of drawing any card from a specific suit, like Spades)
- Option 4: \(\frac{1}{8}\) (This value is not directly related to this problem's standard probabilities)
Our calculated probability, \(\frac{1}{{52}}\), matches Option 1.
| Event | Favorable Outcomes | Total Outcomes | Probability |
|---|---|---|---|
| Drawing the Queen of Spade | 1 (specifically the Queen of Spade card) | 52 (total cards) | \(\frac{1}{52}\) |
Revision Table: Probability of Card Events
| Event Description | Number of Favorable Outcomes | Total Outcomes | Probability |
|---|---|---|---|
| Drawing any specific card (e.g., 7 of Hearts) | 1 | 52 | \(\frac{1}{52}\) |
| Drawing any Queen | 4 (one Queen in each suit) | 52 | \(\frac{4}{52} = \frac{1}{13}\) |
| Drawing any Spade | 13 (all cards in the Spade suit) | 52 | \(\frac{13}{52} = \frac{1}{4}\) |
| Drawing a Red card | 26 (Hearts and Diamonds) | 52 | \(\frac{26}{52} = \frac{1}{2}\) |
Additional Information on Probability Basics
Probability is a fundamental concept in mathematics that quantifies the likelihood of events. It is always a value between 0 and 1, inclusive. A probability of 0 means the event is impossible, and a probability of 1 means the event is certain to occur.
- Sample Space: The set of all possible outcomes of a random experiment. In this card problem, the sample space is the set of all 52 cards.
- Event: A subset of the sample space. Drawing the queen of spade is one specific event.
- Mutually Exclusive Events: Events that cannot occur at the same time. Drawing a Queen of Spades and drawing a King of Hearts are mutually exclusive.
- Independent Events: Events where the outcome of one does not affect the outcome of the other. If you draw a card, replace it, and then draw another, the draws are independent.
Understanding these basic terms helps in solving more complex probability problems involving decks of cards or other random experiments.
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