Understanding the Probability of Rolling a Sum of 7 Given the First Die Shows a 3

The question posed is: if two fair six-sided dice are rolled one after the other, and given that the first die shows a 3, what is the probability that their sum will be 7? This problem isn't just a theoretical curiosity; grasping its solution is crucial for improving your understanding of probability and its applications. Let's dive into the detailed explanation.

Step-by-Step Analysis

To solve the problem, let's break it down step-by-step. It's essential to follow a logical sequence that considers all relevant factors.

Identify the Outcome of the First Die

The first die shows a 3. This is our given condition, and it will influence the possible outcomes for the second die.

Determine What the Second Die Needs to Show

For the sum of the two dice to be 7, the second die must:

[7 - 3 4]

The second die needs to show a 4 to achieve a sum of 7.

Possible Outcomes for the Second Die

The second die can roll any number from 1 to 6. This means there are six possible outcomes for the roll of the second die: 1, 2, 3, 4, 5, and 6.

Calculate the Probability

Since only one favorable outcome (4) can lead to a sum of 7, and there are six possible outcomes for the second die, the probability can be calculated as:

[P(text{sum} 7 mid text{first die} 3) frac{text{Number of favorable outcomes}}{text{Total outcomes}} frac{1}{6}]

Therefore, the probability that the sum of the two dice is 7 given that the first die shows a 3 is (frac{1}{6}).

Visualizing the Problem

Another way to understand this problem is by visualizing it using a 6 x 6 matrix. Label the columns and rows from 1 to 6, and in each cell, write the sum of the corresponding column and row labels. This matrix will help you count the number of cells that have a sum of 7. Here’s a simplified version:

1 2 3 4 5 6 1 2 3 4 5 6 2 3 4 5 6 7 3 4 5 6 7 8 4 5 6 7 8 9 5 6 7 8 9 10 6 7 8 9 10 11

In this matrix, there are six cells that sum to 7 (highlighted with a specific shade if visible). Each row corresponds to one of the six possible outcomes of the first die. The probability of rolling a 7 is thus (frac{6}{36} frac{1}{6}).

Conclusion

While it might seem counterintuitive that the outcome of the first die doesn't change the probability, understanding these principles can be crucial for more complex probability problems. Remember, the basic rule here is that the probability of rolling a specific number on the second die remains (frac{1}{6}).

Probability is a fundamental concept in mathematics and statistics, with wide-ranging applications in fields such as finance, data analysis, and even everyday decision-making. Mastering these concepts can greatly enhance your analytical skills and problem-solving abilities.