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.
Important Questions from Probability of Random Experiments
Two cards are drawn successively without replacement from a well-shuffled pack of 52 cards. The probability of drawing two aces is
A biased coin with the probability of getting head equal to \(\frac{1}{4}\) is tossed five times. What is the probability of getting tail in all the first four tosses followed by head ?
Three dice are thrown. What is the probability that each face shows only multiples of 3 ?
A natural number n is chosen from the first 50 natural numbers. What is the probability that \(n+\frac{50}{n}<50 \) ?
If two dice are thrown, then what is the probability that the sum on the two faces is greater than or equal to 4?
