The ability to systematically list and organize data is a fundamental skill that helps us make sense of complex information. Whether it's generating combinations, permutations, or organizing data sets, systematic listing provides a structured approach to analyze and understand patterns. In this article, we will delve into the world of systematic listing, exploring its definition, various methods, and the underlying principles that drive its effectiveness.

## Systematic Listing of Outcomes Explanation

**Systematic listing of outcomes** is the process of methodically listing all of the possible outcomes of an event in a way that ensures that no outcome is missed out.

Systematic listing of outcomes allows us to calculate the probability of an event occurring, as all of the possible outcomes are listed. This means that the probability of an event occurring is the number of times that event appears in the listing of outcomes divided by the total number of outcomes. However, this can only be done if the probability of each individual event is equal, for example, if an unbiased coin is flipped or an unbiased dice is rolled.

## Systematic Listing of Outcomes Method

Systematic listing of outcomes can be done by inspection. This means that using the information from the situation, you decide which way is the best way to systematically list the possible outcomes. Let's take a look at an example to see how this is generally done:

**Szymon is at a restaurant. He orders a three-course meal. The options for each course are as follows:**

**Starter: Soup, Breadsticks**

**Main: Pizza, Burger**

**Dessert: Ice Cream, Fruit Salad**

**List all of the possible meals that Szymon could order.**

**Solution:**

A good way to systematically list outcomes is to start by making all but one of the options fixed and list all of the outcomes that can come as a result of it. For example, we can start by listing all of the possible meals that include soup as the starter and pizza as the main. This gives us:

Soup, Pizza, Ice Cream

Soup, Pizza, Fruit Salad

Next, we can change the main to burger, giving us:

Soup, Burger, Ice Cream

Soup, Burger, Fruit Salad

Now we can repeat the process but with breadsticks as the starter.

Breadsticks, Pizza, Ice Cream

Breadsticks, Pizza, Fruit Salad

Breadsticks, Burger, Ice Cream

Breadsticks, Burger, Fruit Salad

This method of listing outcomes is known as the **fundamental principle of systematic listing**. It ensures that no outcome is missed out.

Another way of systematically listing outcomes is by using a **sample space diagram.**

A **sample space diagram** is a table that lists all of the possible outcomes of an event that is decided by a combination of two separate events.

Sample space diagrams are a useful tool for visualizing the outcomes of two events. To create a sample space diagram, start by creating a table with the outcomes of the first event as column headers and the outcomes of the second event as row headers. Then, fill in each box with the result of the corresponding headers.

These diagrams are particularly helpful when calculating probabilities for events involving two outcomes. For example, if you were to spin two spinners with numerical values and add the results together, a sample space diagram could help you visualize all possible outcomes.

When calculating the number of outcomes, you can:

- Count the number of squares containing the desired outcome.
- Multiply the number of rows by the number of columns.
- Divide the total number of outcomes by the number of desired outcomes.

**When two six-sided dice are rolled, the numbers obtained from each dice roll are added together. Here is a sample space diagram displaying all possible outcomes:**

**Solution:**

Each dice has numbers from 1 to 6. We can list the outcomes in a table:

1 | 2 | 3 | 4 | 5 | 6 | |

1 | 2 | 3 | 4 | 5 | 6 | 7 |

2 | 3 | 4 | 5 | 6 | 7 | 8 |

3 | 4 | 5 | 6 | 7 | 8 | 9 |

4 | 5 | 6 | 7 | 8 | 9 | 10 |

5 | 6 | 7 | 8 | 9 | 10 | 11 |

6 | 7 | 8 | 9 | 10 | 11 | 12 |

By adding the numbers from the column and row headings, we get the following results:

1 | 2 | 3 | 4 | 5 | 6 | |

1 | 2 | 3 | 4 | 5 | 6 | 7 |

2 | 3 | 4 | 5 | 6 | 7 | 8 |

3 | 4 | 5 | 6 | 7 | 8 | 9 |

4 | 5 | 6 | 7 | 8 | 9 | 10 |

5 | 6 | 7 | 8 | 9 | 10 | 11 |

6 | 7 | 8 | 9 | 10 | 11 | 12 |

## Examples of Systematic Listing of Outcomes

Systematic listing of outcomes is a useful method when dealing with events that have a large number of possible outcomes. It helps in listing all the possible outcomes systematically. Let's take a look at an example to understand this concept better.

**Example:** Two three-sided spinners containing the numbers 1, 2, and 3 are rolled, and the result of each spin is recorded to form a 2-digit number. What are the possible numbers that can be made?

**Solution:**

In this scenario, there are 2 digits in the final number, each with 6 possible values. To list all the possible numbers systematically, we can start by listing the outcomes for the first digit:

1: 11, 12, 13

2: 21, 22, 23

3: 31, 32, 33

By following this systematic approach, we can ensure that all possible outcomes are accounted for.

**When two six-sided dice are rolled, what is the probability that the sum of the dice rolls is 7?**

**Solution:**

When rolling two dice, each dice has 6 possible outcomes. To find the probability of getting a sum of 7, we need to determine the number of ways the dice rolls can add up to 7.

A table listing the results of each dice roll reveals 6 combinations that total 7:

1 | 2 | 3 | 4 | 5 | 6 |

1 | 6 | 5 | 4 | 3 | 2 |

There are a total of 36 possible outcomes when rolling two dice (6 outcomes for the first dice and 6 outcomes for the second dice).

Therefore, the probability of getting a sum of 7 is $\frac{6}{36}$, which simplifies to $\frac{1}{6}$.

## The Importance of Systematic Listing of Outcomes

Systematic listing of outcomes is crucial in ensuring accuracy and efficiency in the process of listing the outcomes of events. By methodically listing all possible outcomes, we can avoid missing any important outcomes and save time in the process. This systematic approach becomes even more essential when dealing with events that have a large number of possible outcomes.

## Conclusion

Systematic listing is a powerful technique that facilitates organized data representation and analysis. Through the use of structured methods and principles, it enables us to efficiently generate combinations, permutations, and organized data sets. By exploring the various methods and understanding the underlying principles of systematic listing, we gain a versatile toolset for problem-solving, pattern recognition, and data organization.