Problem 53 - 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 the marble is red or yellow?

Step-by-Step Solution

Answer

The probability that the marble is red or yellow is 0.73.

Step 1: Identify Total Possible Outcomes

Determine the total number of marbles in the bag. Sum up all the marbles, which is \(5 + 1 + 3 + 2 = 11\). So, there are 11 marbles in Bag 1.

Step 2: Identify Successful Outcomes

Determine the number of marbles that are red or yellow. Sum up the red and yellow marbles, which is \(5 + 3 = 8\). So, these are 8 successful outcomes.

Step 3: Calculate Probability

To find the probability of the marble drawn being red or yellow, divide the number of successful outcomes by the total number of outcomes. Hence the probability is \(\frac{8}{11} = 0.73\).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of objects. It's essential in probability when determining all possible outcomes and events. In our marble problem, combinatorics helps us understand how many different ways we can draw marbles from the bag.
For example, by listing out all marbles, we're using combinatorics to see how many total marbles there are. In Bag 1, there are a total of 11 marbles, made up of different colors. Combinatorics also aids in figuring out how many favorable outcomes, i.e., drawing either a red or a yellow marble, add up to 8 out of these 11 possibilities.

• Understand the total set - all marbles in the bag.
• Determine subsets - marbles of specific colors.

This basic concept helps build the foundation for calculating probability. It involves skills like counting and listing elements systematically, which are crucial in more complex problems.

Addition principle

The addition principle in probability or combinatorics guides us in situations where we need to choose between different possibilities. It states that if two events are mutually exclusive, the total number of successful outcomes is found by adding together the number of ways each event can occur.
In the context of the marble problem, we have two separate events - drawing a red marble or a yellow marble. Since a single marble cannot be both red and yellow simultaneously, these events are mutually exclusive. Thus, by the addition principle:

• Count red outcomes (5 marbles).
• Count yellow outcomes (3 marbles).
• Add them to find the total successful outcomes (5 + 3 = 8).

This principle is handy when calculating total probabilities from multiple events, ensuring you correctly account for all favorable cases without overlap, especially when events cannot occur at the same time.

Marble probability

Probability gives us a way to measure the likelihood of a specific event happening. It can be expressed as a fraction, a percentage, or a decimal. In our example, we're focusing on calculating the probability of drawing a red or yellow marble from Bag 1.
To find this probability, we follow a straightforward process:

• Identify the total number of marbles (11).
• Count the red and yellow marbles, which represent our successful outcomes (8).
• Divide the number of successful outcomes by the total number of outcomes to find the probability: \(\frac{8}{11}\).

Converting it to a decimal gives an easy-to-understand representation \(0.73\), meaning there's a 73% chance of drawing a red or yellow marble. This concept is key in real-world applications where making predictions based on evidence is needed, like in games of chance or even weather forecasts.

Fraction conversion

Fraction conversion is the process of changing fractions into decimals or percentages. This is especially useful in probability to aid understanding and communication. It's much easier to grasp and compare numbers when they're in the same format.
For the probability of drawing a red or yellow marble, the fraction \(\frac{8}{11}\) can be converted into a decimal by dividing the numerator by the denominator, resulting in \(0.73\).
This can further be expressed as a percentage by multiplying by 100, giving us \(73\%\).

• Fractions provide a precise representation.
• Decimals offer an easy-to-read format.
• Percentages make comparisons intuitive, like understanding likelihood.

Conversion improves the clarity of probability statements, making statistics more accessible and actionable in everyday decisions and academic contexts.