Problem 48 - Algebra 2

Question

For Exercises \(46-48\) , use the following information. A bag contains 10 marbles. In this problem, a cycle means that you draw a marble, record its color, and put it back. How many times do you have to repeat the cycle to be certain that there are no black marbles in the bag? Explain your reasoning.

Step-by-Step Solution

Answer

Repeat the cycle 10 times to be certain there are no black marbles.

Step 1: Understand the Problem

We are tasked to determine how many times we must draw marbles, record their color, and return them to the bag to be certain no black marbles are present in the bag.

Step 2: Establish the Worst Case Scenario

In the worst-case scenario, we could keep drawing and not find any black marbles, even if they are present. We must ensure that our draws cover all possibilities.

Step 3: Determine the Total Number of Marbles

The bag contains 10 marbles in total, as given in the problem.

Step 4: Calculate Draws Needed to Ensure Drawing Each Type of Marble

To be certain there is no black marble, we must draw all 10 marbles, to test each one, assuming the worst case scenario where black is the last color to be seen if present.

Step 5: Conclude the Required Number of Cycles

We conclude that we must repeat the cycle 10 times to be certain that each marble has been drawn at least once and thus, verify there are no black marbles.

Key Concepts

Random Sampling

Random sampling is a crucial concept in probability theory and statistics. It's a method we use to select a random and unbiased sample from a larger population. In the context of the marble problem, random sampling refers to the process of drawing a marble from the bag multiple times.
Each draw is an independent event, and the result of each draw doesn't affect the others. When we randomly sample, we aim to get a representative sample of the population—in this case, the marbles in the bag.

• Each draw gives us information about the presence or absence of black marbles.
• The randomness ensures that every marble in the bag has an equal chance of being selected in each draw.
• Random sampling helps in making statistical inferences about the whole bag.

It's important to understand that while each draw is random, the more draws we make, the more confident we can be about our conclusions regarding the bag's contents.

Worst-Case Scenario

A worst-case scenario is a concept often used to evaluate the most unfavorable outcome in a given situation. In our marble problem, we need to consider the worst-case scenario to ensure that our conclusion about there being no black marbles is absolutely certain.

• In the context of our problem, the worst-case scenario occurs when we have to draw every single non-black marble before finally confirming the absence of black marbles.
• This scenario accounts for the possibility that the black marble, if it exists, could be the very last marble drawn.
• Analyzing the worst-case scenario ensures we have thoroughly covered all potential outcomes in our inquiry.

Thinking in terms of the worst-case helps us prepare for any possibility, ensuring our strategy is foolproof.

Replacement Sampling

Replacement sampling, also known as sampling with replacement, is a method where each sampled unit is returned to the population before the next draw. In our marble problem, this means that after each draw, the marble is placed back into the bag, keeping the total number of marbles the same for every cycle.
This type of sampling affects our probability calculations in a few important ways:

• Each draw remains independent, as the probability of drawing any specific marble remains constant after each cycle.
• Replacement helps maintain the same conditions for each sample, allowing us to repeat the process without introducing biases.
• It makes sure our sampling model reflects the idea of infinite potential repetitions, allowing us to address the problem in probabilistic terms rather than deterministic ones.

Understanding replacement sampling is key to solving the problem because it ensures that every draw remains equally likely and unbiased, making our findings reliable.