Understanding Probability in Pen Selection: A Comprehensive Guide

When dealing with probability problems, it's essential to break down the problem step by step to understand the underlying principles. In this article, we will explore the probability of selecting a blue pen and a red pen from a pen holder containing various pens, all while understanding the importance of replacing the selected pens.

The Problem and Its Components

Suppose you have the following pens in a pen holder:

5 purple pens 4 blue pens 3 red pens 6 black pens

The total number of pens in the pen holder is:

5   4   3   6  18 pens

Calculating the Probability

The probability of an event is calculated as the number of favorable outcomes divided by the total number of possible outcomes. Let's break down the problem into smaller parts and solve it step by step.

Step 1: Probability of Selecting a Blue Pen First

The total number of blue pens is 4, and the total number of pens is 18. Therefore, the probability of selecting a blue pen on the first draw is:

Probability of selecting a blue pen  4 / 18  2 / 9

Step 2: Probability of Selecting a Red Pen Second

Since the pen is replaced after the first draw, the total number of pens remains the same at 18. The number of red pens is 3, so the probability of selecting a red pen on the second draw is:

Probability of selecting a red pen  3 / 18  1 / 6

Step 3: Using the Multiplication Rule

The multiplication rule for independent events states that the probability of two independent events occurring is the product of their individual probabilities. In this case, the events are independent because the pen is replaced after the first draw. Therefore:

Probability of selecting a blue pen and a red pen  Probability of selecting a blue pen × Probability of selecting a red pen

Substituting the values we have:

Probability  (2 / 9) × (1 / 6)  2 / 54  1 / 27

Therefore, the probability of selecting a blue pen and a red pen is 1/27.

Conclusion

By understanding the basic principles of probability and the concept of independent events, we can solve complex problems such as the pen selection scenario. The key to solving such problems is breaking them down into smaller, manageable steps and ensuring that each step is accurately calculated.

For more detailed analysis and additional examples, refer to the resources below:

Probability Basics - Math is Fun Khan Academy Probability and Statistics

Remember, the phrase “pens are replaced” is crucial in determining the probability. If the pens are not replaced, the second probability would adjust accordingly to account for the reduced number of pens in the pen holder.