Problem 55 - Algebra 2

Question

Use this information for Exercises \(53-58\) . Bag 1 contains 5 red marbles, 1 blue marble, 3 yellow marbles, and 2 green marbles. Bag 2 contains 1 red pencil, 3 red pens, 2 blue pencils, and 5 blue pens. One marble is drawn from bag \(1 .\) What is the probability that it is blue or yellow or green?

Step-by-Step Solution

Answer

The combined probability is \(\frac{6}{11}\).

Step 1: Calculate Total Marbles in Bag 1

Add up all of the marbles in Bag 1. Red marbles (5), blue marble (1), yellow marbles (3), and green marbles (2) equates to eleven total marbles.

Step 2: Calculate Individual Probabilities

Next, figure out the probability of drawing a blue, yellow, or green marble from Bag 1. Repeat this step for each type of marble. The probability is calculated by dividing the number of each respective marble color by the total number of marbles. Probability of blue is \(\frac{1}{11}\), the probability of yellow is \(\frac{3}{11}\), and the probability of green is \(\frac{2}{11}\).

Step 3: Calculate Combined Probability

Finally, add together the individual probabilities of drawing a blue, yellow, or green marble to get the combined probability. When you add the probabilities calculated in Step 2, \( \frac{1}{11} + \frac{3}{11} + \frac{2}{11} = \frac{6}{11}\).

Key Concepts

Combinatorics

Combinatorics is a fascinating field of mathematics that deals with counting, arranging, and combining objects or sets. It's like solving puzzles using logical steps. In this exercise, combinatorics helps us understand how many ways we can choose and categorize marbles from Bag 1. When you're asked about the probability of drawing a blue, yellow, or green marble, combinatorics makes it easier to visualize and solve the problem by considering each subset of marbles and their contributions to the total possibilities:

• Understand grouping: You categorize the marbles into groups (red, blue, yellow, green) to determine potential outcomes.
• Identify desired outcomes: You're interested in groups that influence the probability of events (drawing blue, yellow, or green marbles).
• Calculate possibilities: Evaluate the possible outcomes to assess the probability.

Combinatorics provides a structured way to break down complex problems into manageable parts. This method supports finding solutions efficiently without overlooking any potential scenarios.

Counting

Counting is a foundational skill in mathematics. It ensures you know exactly the number of items or outcomes available, which is crucial when determining probabilities. In the exercise, counting helped establish the total number of marbles in Bag 1. Here’s how it works: Firstly, you take stock of every type of marble in the bag. By summing these numbers, you discovered a total of eleven marbles.

• 5 red marbles
• 1 blue marble
• 3 yellow marbles
• 2 green marbles

This count provides a baseline for subsequent calculations. When you want to know the probability of choosing a blue, yellow, or green marble, you must first know how many of those marbles exist. Counting ensures your probability calculations are based on accurate figures. Being thorough with counting at the beginning of any exercise eliminates errors and lays a solid foundation for understanding and solving the problem.

Mathematical reasoning

Mathematical reasoning involves using logical thought processes to solve problems. It’s like connecting pieces of a puzzle to see the big picture. In probability exercises like this one, reasoning guides us through the order of steps to reach an answer.To solve the problem of finding the probability of picking a blue, yellow, or green marble:

• Start by confirming the definite count of total marbles (11 marbles), setting the stage for accurate calculations.
• Determine probabilities for each event (picking a blue, yellow, or green marble) by dividing their counts by the total number of marbles.
• Use addition to solve for the combined probability of multiple independent events happening (blue or yellow or green).

Logical reasoning here dictates that each probability builds upon knowing the total marble count and individual color counts. Through each logical step, mathematical reasoning helps you find and understand how the \[\frac{6}{11}\] result emerges, ensuring clarity of thought and awareness of each step's influence in the problem-solving process.