Math Problem Statement

Ron randomly pulls a pen out of a box that contains 1 red, 2 black, and 3 blue pens. He does this 4 times, replacing the pen each time, but pulls out a blue pen only 1 time. Ron concludes that the observed frequency of pulling a blue pen will eventually be closer to the expected frequency based on the theoretical probability of pulling a blue pen. Which reasoning best supports his conclusion?

Solution

To analyze this problem, let’s first break down the details provided:

  1. Pen Composition: The box has 1 red pen, 2 black pens, and 3 blue pens. This makes a total of 1+2+3=61 + 2 + 3 = 6 pens.

  2. Theoretical Probability of Blue Pen: The probability of pulling a blue pen from the box is calculated as: number of blue penstotal number of pens=36=12\frac{\text{number of blue pens}}{\text{total number of pens}} = \frac{3}{6} = \frac{1}{2}

  3. Experimental Observation: Ron pulls a pen out 4 times and only gets a blue pen once. This observed frequency suggests that in his experiment, the chance of pulling a blue pen seems lower than the theoretical probability.

  4. Law of Large Numbers: Ron believes that as he increases the number of trials (pulling pens), the observed frequency (experimental probability) of pulling a blue pen will get closer to the theoretical probability.

Given the answer choices:

  • The correct answer should reflect that the experimental probability will get closer to the theoretical probability (1/2) as the number of trials increases. This is based on the principle of the Law of Large Numbers, which states that the more trials conducted, the closer the experimental probability will align with the theoretical probability.

Therefore, the correct answer is: "The experimental probability of pulling a blue pen based on the observed frequency will get closer to 12\frac{1}{2} as the number of trials increases."

Would you like further explanation on probability concepts, or do you have additional questions?

Here are five related questions that might help you explore this topic further:

  1. What is the difference between theoretical and experimental probability?
  2. How does the Law of Large Numbers influence probability experiments?
  3. Why might experimental probability differ from theoretical probability in a small sample size?
  4. How would increasing the number of trials affect the accuracy of experimental probability?
  5. Can the experimental probability ever be exactly equal to the theoretical probability?

Tip: Remember that experimental probability often fluctuates with a smaller number of trials, but it generally stabilizes closer to the theoretical probability as trials increase.

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Math Problem Analysis

Mathematical Concepts

Probability
Experimental Probability
Theoretical Probability
Law of Large Numbers

Formulas

Theoretical Probability = (Number of favorable outcomes) / (Total number of outcomes)

Theorems

Law of Large Numbers

Suitable Grade Level

Grade 7-9