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Probability is a way to measure the likelihood of an event happening. It's calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In our example with the balls, we wanted to find the probability of drawing a black ball and a non-red ball from a box containing various colored balls.

To calculate the probabilities, follow these steps:

• First, identify the total number of outcomes. In this case, it is the total number of balls in the box, which is 9.
• Next, figure out the favorable outcomes for the specific event. For drawing a black ball, there are 3 black balls, so there are 3 favorable outcomes.
• Lastly, use the fraction \((\frac{Number\; of\; Favorable\; Outcomes}{Total\; Number\; of\; Outcomes})\) to find the probability. For a black ball: \(\frac{3}{9} = \frac{1}{3}\), and for a non-red ball: \(\frac{5}{9}\).

Understanding how to calculate probabilities will help you with many other probability-related problems.

A random variable is a variable that can take on different values, each with a certain probability. In the context of our problem, the random variable is the color of the ball that is drawn from the box. Since the ball is drawn at random, every ball has an equal chance of being picked.

When we say we are drawing a ball at random, it means each ball - regardless of its color - has an equal chance of being chosen. Here, each of the 9 balls has an equal 1 in 9 chance initially. After we identify the color categories, like black or non-red, we can calculate the specific probability based on these categories.

Favorable outcomes are the specific outcomes that we are interested in for a given event. In probability, these are the successful outcomes that contribute to the event happening.

For the event of drawing a black ball, the favorable outcomes are the counts of black balls in the box. In our example, there are 3 black balls, so there are 3 favorable outcomes for drawing a black ball.
For the event of drawing a non-red ball, the favorable outcomes are the counts of all balls that are not red. With 2 white and 3 black balls, there are 5 non-red balls, thus 5 favorable outcomes for drawing a non-red ball.

To summarize:

• Favorable outcomes for drawing a black ball: 3 (the number of black balls)
• Favorable outcomes for drawing a non-red ball: 5 (the sum of white and black balls)

Knowing the favorable outcomes and the total number of possible outcomes allows us to find probabilities easily.