Two Red Balls from Bag

Sampling Without Replacement

When items are selected from a finite set without replacement, each selection changes the composition of the remaining set. This means the probability of subsequent selections depends on the outcomes of previous selections. These are called dependent events.

For example, if you draw a red ball from a bag and don't put it back, there's one fewer red ball and one fewer total ball for the next draw. This is a common scenario in many real-world probability problems, from card games to quality control sampling.

Approaches to Calculation

There are generally two main ways to calculate probabilities in such scenarios:

1. Sequential Approach: Calculate the probability of each step in the sequence, considering how the previous steps affect the current one (using conditional probability), and then multiply these probabilities.
2. Combinatorial Approach: Use combinations to count the number of ways to achieve the desired outcome (e.g., selecting 2 red balls) and divide it by the total number of ways to make the selections (e.g., selecting any 2 balls). This method is useful when the order of selection doesn't matter for the final outcome.

Both methods should yield the same result if applied correctly.