Probability of Drawing Two Black Cards from a Standard Deck Without Replacement

A standard deck of 52 cards consists of 26 black cards, specifically the spades and clubs. This article will guide you through the process of calculating the probability of drawing two black cards consecutively, without replacing the first card.

Understanding the Basics

When dealing with a probability problem involving cards, it is important to break the problem down into smaller, manageable steps. This allows for a more accurate calculation and better understanding of the underlying principles. The problem at hand requires us to calculate the probability of two events occurring in succession:

First, drawing a black card. Second, drawing another black card without replacing the first one.

Calculating the Probability

Step 1: Probability of Drawing the First Black Card

The deck starts with 52 cards, and half of them are black. Therefore, the probability of drawing a black card on the first draw is:

[P_{text{first black}} frac{26}{52} frac{1}{2}]

Step 2: Probability of Drawing the Second Black Card

Without replacing the first card drawn, we are now dealing with a deck of 51 cards, 25 of which are black. Therefore, the probability of drawing a second black card is:

[P_{text{second black | first black}} frac{25}{51}]

Step 3: Combined Probability

To find the combined probability of both events occurring, we multiply the probabilities of each event:

[P_{text{both black}} P_{text{first black}} times P_{text{second black | first black}} frac{1}{2} times frac{25}{51} frac{25}{102}]

Conclusion

The probability that two cards drawn from a standard deck of 52 cards, without replacement, are both black is (frac{25}{102}), which is approximately 24.51%. This calculation is crucial for understanding basic probability concepts and can be applied to various scenarios involving card draws.

Related Keywords and Concepts

Probability Calculation: The process of determining the likelihood of an event occurring based on mathematical principles. Deck of Cards: A standard 52-card deck, consisting of four suits (spades, hearts, diamonds, clubs). Without Replacement: A sampling method where the selected items are not put back into the pool, affecting the probability of subsequent draws.

By mastering these concepts, you can apply them to more complex probability problems and deepen your understanding of statistical probability.