Problem 57 - 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 a pen or a red pencil?

Step-by-Step Solution

Answer

The probability of drawing a pen or a red pencil from bag 2 is \(\frac{9}{11}\).

Step 1: Identify Total Number of Items in Bag 2

First of all, we need the total number of items in bag 2. We do so by adding all the items together which are 1 red pencil, 3 red pens, 2 blue pencils, and 5 blue pens. This results in 11 items in total \((1+3+2+5=11)\).

Step 2: Identify Desired Outcomes

Our desired outcomes are defined by the case that we draw a pen or a red pencil from the bag. We add up the total number of pens and red pencils to get the total number of desired outcomes. This results in 9 desired outcomes \((3 red pens + 5 blue pens + 1 red pencil = 9)\).

Step 3: Calculate the Probability

To calculate the probability, we need to divide the number of desired outcomes by the total number of outcomes: \(P = \frac{Number\:of\:Desired\:Outcomes}{Total\:Number\:of\:Outcomes} = \frac{9}{11}\)

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and analyzing sets. When dealing with problems like the one described with marbles and pencils, combinatorics helps us determine the number of possible outcomes. For instance, in determining how many items we have in Bag 2, we use combinatorial counting: 1 red pencil, 3 red pens, 2 blue pencils, and 5 blue pens add up to a total of 11 items.
This foundational counting is the basis for calculating probabilities since knowing all possible outcomes (or total samples) allows for determining the likelihood of specific outcomes.

Sample Space

The sample space in probability refers to the set of all possible outcomes of an experiment. In our exercise, the sample space consists of drawing one item from Bag 2, which contains 11 items in total. These include one red pencil, three red pens, two blue pencils, and five blue pens.
When computing probabilities, the sample space – in this case, 11 items – serves as the denominator in our probability fraction. It represents all the potential outcomes against which desirable outcomes are measured.

Desirable Outcomes

Desirable outcomes are the specific events or results we are interested in when calculating probabilities. In the exercise provided, we want to find the probability of drawing either a pen or a red pencil from Bag 2. These desirable outcomes include:

• Red pens: 3
• Blue pens: 5
• Red pencil: 1

Adding these together, we get 9 desirable outcomes because pens and the red pencil satisfy our condition.
These outcomes are used to compare against the sample space to determine probability, serving as the numerator in our probability equation.

Independent Events

Independent events are those where the outcome of one event does not affect the outcome of another. In the context of this problem, drawing items from Bag 2 can be considered independent of drawing items from Bag 1. This means that the composition and the drawing procedure in Bag 1 have no bearing on the likelihood of drawing a specific item from Bag 2.
Understanding the notion of independent events can simplify the problem. In this case, determining the probability of drawing a pen or red pencil involves only evaluating Bag 2 since Bag 1's contents don't influence Bag 2's draws. This separation of probabilities maintains clarity in calculation and ensures accuracy in results.