91Ó°ÊÓ

Understanding independent events is crucial when considering probability tasks, such as drawing marbles from a box. An event is considered independent when the outcome of one event does not affect the outcome of another event. In our exercise, because we draw each marble with replacement, the two draws are independent.

This means the color of the first marble drawn has no effect on the color of the second marble drawn.

• Each time we draw a marble, the set of marbles remains unchanged.
• The probability of drawing any color remains the same across all draws.

This independence allows us to calculate the combined probability by simply multiplying the probabilities of the individual events. When considering the sample space, each draw is treated separately.

The term 'with replacement' is significant because it specifies that once a marble is drawn, it is placed back into the box before the next draw. This action keeps the total number of marbles the same and preserves the original probabilities.

Using our exercise as an example, we have:

• 8 red marbles,
• 8 yellow marbles,
• 8 green marbles.

When drawing two marbles with replacement, each draw is an independent event. Therefore, after drawing one marble, there are still 8 marbles of each color left in the box for the next draw.

• This maintains a consistent probability for each marble color.
• It simplifies the process of calculating the overall probabilities.

Understanding replacement is key to building accurate probability models.

Probability outcomes refer to the possible results of an experiment. In terms of the marble exercise, the outcomes are determined by the combination of colors drawn. When you draw marbles with replacement, and each draw is independent, the number of outcomes can be calculated using the formula: \[ \text{Total outcomes} = \text{Outcomes per draw} \times \text{Number of draws} \] In this exercise:

• 3 potential outcomes per draw (Red, Yellow, Green)
• 2 draws

Thus, we have: \(3 \times 3 = 9\) outcomes. These outcomes are represented in pairs such as (R, R) or (Y, G).

By listing every possible pair, we create a sample space, which includes all potential outcomes for the experiment. Recognizing and recognizing probability outcomes allows for a better understanding of the likelihood of each potential result.