Question

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?

Short answer

The probability of drawing a black ball is \( \frac{1}{3} \) and the probability of drawing a non-red ball is \( \frac{5}{9} \).

Step-by-step solution

- Total Number of Balls

Count the total number of balls in the box. There are 2 white, 3 black, and 4 red balls. So, the total number of balls is: \[ 2 + 3 + 4 = 9 \]

- Number of Black Balls

Identify the number of black balls. From the problem, there are 3 black balls.

- Probability of Drawing a Black Ball

Use the formula for probability, which is the number of favorable outcomes divided by the total number of outcomes. Therefore, the probability that a ball drawn is black is: \[ \frac{3}{9} = \frac{1}{3} \]

- Number of Non-Red Balls

Calculate the number of non-red balls by subtracting the number of red balls from the total number of balls. There are 4 red balls in a total of 9 balls, so the number of non-red balls is: \[ 9 - 4 = 5 \]

- Probability of Drawing a Non-Red Ball

Again use the probability formula. The probability that a ball drawn is not red is: \[ \frac{5}{9} \]

Key concepts

Probability Calculation

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.

Random Variable

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

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.