In a box there are 2 white, 3 black, and 4 red balls. If a ball is drawn at random, what is the probability that it is black? That it is not red?

Probability calculation is a fundamental aspect of probability theory that helps us understand the likelihood of events occurring. In this exercise, we calculated the probability of drawing a black ball and a non-red ball from a box containing white, black, and red balls.
To do this, follow these steps:
1. Count the total number of elements (in this case, balls) in the set.
2. Identify the subset of elements representing the event of interest (e.g., black balls or non-red balls).
3. Calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes.

For example, the probability of drawing a black ball was calculated as follows: \( P(\text{Black}) = \frac{3}{9} = \frac{1}{3} \)
Similarly, the probability of drawing a ball that is not red was calculated as: \( P(\text{Not Red}) = \frac{5}{9} \)

Always remember to simplify fractions whenever possible for a clearer and more concise result.

Combinatorics is the branch of mathematics dealing with combinations, permutations, and counting. It is essential in probability theory to help determine the number of possible outcomes and favorable outcomes.

Here are a few key concepts in combinatorics:

• **Combination**: Selection of items where order does not matter.
• **Permutation**: Arrangement of items where order does matter.
• **Counting Principle**: Using multiplication to find the number of combinations.

In this exercise, we are not explicitly using combinatorics. However, understanding how to count the total number of balls (9 in this case) and the number of black balls (3) or non-red balls (5) relies on basic counting principles.

Remember, always ensure accurate counting of the entire set and subsets representing the events of interest to accurately apply probability calculations.

Basic statistics involves the collection, analysis, interpretation, and presentation of data. In the context of this exercise, we used simple statistical measures to calculate probabilities.

Understanding the idea of 'proportion' is crucial. A proportion represents the fraction of a subset within the entire set.

• Calculation: Proportion = \(\frac{Number \ of \ subset \ elements}{Total \ number \ of \ elements} \)
• Presentation: Simplify the fraction for clarity.

For example, the proportion of black balls is \(\frac{3}{9}\), which simplifies to \(\frac{1}{3}\). This expresses that one third of the total balls are black.

Similarly, the probability of drawing a ball that is not red is calculated by determining the number of non-red balls and then forming a proportion: \(\frac{5}{9}\).

Therefore, understanding and using basic statistical methods help in deriving meaningful conclusions from data, especially when calculating probabilities.

Random events are those that occur without a specific pattern or predictability. In probability theory, we often deal with random events to measure their likelihood.

Key points about random events:

• **Independent Events**: Events where the occurrence of one does not affect the other.
• **Dependent Events**: Events where the occurrence of one affects the other.
• **Random Variables**: Numerical outcomes of random events (e.g., drawing a black ball).

In this exercise, drawing a ball from the box is a random event since each ball has an equal chance of being selected.

The probability of drawing a black ball was calculated by identifying the favorable outcomes (black balls) and comparing them to the total number of possible outcomes (all balls in the box). This simple example demonstrates the fundamental nature of random events and the importance of probability theory in understanding their behavior.

Random events are essential to many areas of study and practical applications, from simple games to complex financial models.