Probability of Drawing a Black Queen from a Standard Deck of Cards

When a card is drawn at random from a standard deck of 52 cards, what is the probability of getting a black queen? This question is straightforward and can be answered using basic principles of probability. We will break down the problem and provide a clear, step-by-step solution.

Understanding the Deck of Cards

A standard deck of cards consists of 52 cards, divided into four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards, including the ranks: 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, and Ace. The hearts and diamonds are red, while the clubs and spades are black.

Identifying the Black Queens

There are two black queens in a standard deck: the Queen of clubs (?) and the Queen of spades (?). These are the only two black queens in the deck.

Calculating the Probability

The probability of an event occurring is calculated using the formula:

P(E) Number of favorable outcomes / Total number of possible outcomes

In this case, the favorable outcomes are the two black queens, and the total number of possible outcomes is the total number of cards in the deck, which is 52.

Step 1: Identify the Favorable Outcomes

There are 2 favorable outcomes: the Queen of clubs and the Queen of spades.

Step 2: Identify the Total Number of Possible Outcomes

The total number of possible outcomes is 52, as there are 52 cards in the deck.

Step 3: Calculate the Probability

Using the formula, we get:

P(Black Queen) 2 / 52 1 / 26

This means that the probability of drawing a black queen from a standard deck of 52 cards is 1/26.

Example with Additional Events

Suppose we also draw a red king from the same deck. Let Q represent the event of drawing a black queen, and K represent the event of drawing a red king. We can use the following formula to find the probability of either event occurring:

P(Q ∪ K) P(Q) P(K) - P(Q ∩ K)

Here, P(Q) 2/52 1/26 and P(K) 2/52 1/26. Since there are no cards that are both black queens and red kings, P(Q ∩ K) 0.

Calculating the Probability

P(Q ∪ K) 1/26 1/26 - 0 2/26 1/13

The probability of either drawing a black queen or a red king is 1/13.

Conclusion

Understanding basic probability concepts is crucial when dealing with problems involving cards and other similar scenarios. In this case, the probability of drawing a black queen from a standard deck of 52 cards is 1/26. By breaking down the problem and using the appropriate formula, we can easily find the solution.

Do you have any questions or need further clarification on this topic? If so, feel free to ask!