Problem 58 - 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 item is drawn from bag \(2 .\) What is the probability that it is red or a pencil?

Step-by-Step Solution

Answer

The probability that an item drawn from bag 2 is either red or a pencil is \(\frac{6}{11}\).

Step 1: Calculate Total Items in Bag 2

Calculate the total items in the bag 2 which is a sum of red pencils, red pens, blue pencils, and blue pens. Which is: \(1+3+2+5 = 11\).

Step 2: Calculate Desired Outcomes

Determine the count of items which are either red or a pencil. We have 1 red pencil, 3 red pens and 2 blue pencils. So, \(1+3+2 = 6\).

Step 3: Calculate the Probability

Probability can be calculated as the ratio of desired outcomes to total outcomes. So, Probability is \(\frac{6}{11}\).

Key Concepts

Counting

Counting is the process of determining the total number of items in a set. In probability, counting is essential because it helps to identify the number of possible outcomes. For the given exercise, counting begins by determining the total number of items in Bag 2.

This includes:

• 1 red pencil
• 3 red pens
• 2 blue pencils
• 5 blue pens

Counting these items gives a total of 11. Understanding how to count properly ensures that you account for each possible outcome, which is a crucial step in probability calculations.

Counting also helps identify desired outcomes. In this context, items that are either red or pencils are:

• 1 red pencil
• 3 red pens
• 2 blue pencils

That's a total of 6 items. Proper counting gives a clear sense of both the total possible and desired outcomes, setting the stage for calculating probability.

Probability Theory

Probability theory is the branch of mathematics that studies the likelihood of events occurring. It is fundamentally concerned with quantifying uncertainty. In our scenario, we want to know the probability that the item drawn from Bag 2 is either red or a pencil.

Probability is generally expressed as a ratio. It's the number of desired outcomes divided by the total number of possible outcomes.

In mathematical terms, the probability is calculated as:\[P(A) = \frac{\text{Number of desired outcomes}}{\text{Total number of outcomes}}\]For Bag 2, there are 11 total items and 6 that satisfy our condition (red or pencil). Therefore, the probability is:\[P(\text{red or pencil}) = \frac{6}{11}\]This simple ratio gives a clear picture of how likely it is that when you draw one item from the bag, it will be either red or a pencil, illustrating the core principles of probability theory.

Problem-Solving

Problem-solving in probability involves identifying and applying the correct strategies to find an answer. It requires breaking down a problem into manageable parts and logically progressing through each step.

To solve a probability problem like the one provided, start by:

• Identifying what you're asked to find—in this case, the probability of drawing an item that's either red or a pencil.
• Understanding the problem’s structure, which involves recognizing the properties of the items in Bag 2.
• Using counting to determine the total and the desired outcomes, leading to a computation of probability using these counts.

Critical thinking and logical reasoning are central to problem-solving in probability. By clearly understanding each component and using basic mathematical operations, you can effectively tackle probability tasks. It's all about dissecting the question, analyzing given data, and methodically working towards a solution.