Question

A box contains 30 balls. Among them 10 are black, 12 are blue and the rest are orange. What is the probability that a ball drawn from the box will not be blue? (1) \(1 / 3\) (2) \(3 / 5\) (3) \(5 / 6\) (4) \(1 / 6\)

Short answer

Answer: \(3/5\)

Step-by-step solution

Find the number of orange balls

First, we need to find out how many orange balls there are. Since there are 30 balls in total and 10 are black and 12 are blue, that leaves the rest of the balls being orange: 30 balls - 10 black balls - 12 blue balls = 8 orange balls.

Calculate the probability of not drawing a blue ball

Now we need to find the probability of not drawing a blue ball. Since there are 12 blue balls, there are a total of 30 - 12 = 18 balls that are either black or orange. Therefore, the probability of not drawing a blue ball is the ratio of non-blue balls to the total number of balls, which is (18 non-blue balls) / (30 total balls). So, we have: \(\frac{18}{30} = \frac{3}{5}\) Thus, the probability of not drawing a blue ball is \(3/5\). So, the correct answer is (2) \(3/5\).

Key concepts

Combinatorics

Combinatorics helps us understand and organize how different items can be combined or grouped in various ways. It's a cornerstone in probability problems, as it allows us to calculate possibilities and occurrences of specific outcomes.
Take a box of 30 balls, for example. When asked to find the probability of drawing a ball that isn't blue, combinatorics shows us how to perceive combinations of non-blue balls from the total collection.

• First, identify the composition of the set. In our exercise, we have 30 balls divided into three colors: black, blue, and orange.
• Calculate how many balls are of each color using subtraction, a simple part of combinatorics.
• Use these groupings to identify possible outcomes, like drawing any ball that isn't blue.

By using combinatorics, you can tabulate the number of ways scenarios like drawing a non-blue ball can happen. This example is straightforward, but combinatorics forms the basis for much more complex probability scenarios.

Basic arithmetic

Basic arithmetic serves as the foundation for solving probability and combinatorial problems. Understanding operations such as addition, subtraction, multiplication, and division are crucial for calculating probabilities.
In our example, arithmetic was employed to determine the number of orange balls in the box. We started with a total and subtracted known quantities (black and blue balls) to find the unknown (orange balls).

• Start with the total: 30 balls.
• Subtract the number of black balls: 30 - 10 = 20.
• Subtract the number of blue balls: 20 - 12 = 8. This leaves us with 8 orange balls.

Finally, basic arithmetic helps us progress to the probability calculation. The total number of non-blue balls (black and orange) is calculated through subtraction, leading us to the next step: assessing the fraction of interest.

Fraction simplification

Fraction simplification makes our answers more digestible and easy to understand. It involves reducing fractions to their simplest form by finding a common divisor.
In probability, once we have the fraction representing an event, simplifying it helps clarify the scale of chance involved.

• Identify the probability fraction: \(\frac{18}{30}\) represents the scenario of not drawing a blue ball, created by dividing non-blue ball counts by the total number of balls.
• Find the greatest common divisor (GCD). Here both 18 and 30 share a GCD of 6.
• Divide both the numerator and the denominator by this GCD:
  \(\frac{18 \div 6}{30 \div 6} = \frac{3}{5}\)

Thus, the probability fraction simplifies to \(\frac{3}{5}\), clearly delineating the chance of drawing a non-blue ball. Simplification not only makes numbers easier to manage but often provides clearer insights into probabilities.