Suppose we have a PIN number for unlocking a mobile phone. If the PIN number is 4 digits long and each individual digit can be any number from 0-9, what is a methodical and efficient way of listing all the possible combinations of available PIN numbers? This article will explain the systematic listing of outcomes which allows us to easily list all of the outcomes of an event.
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 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 created by creating a table, heading the columns with the outcomes of the first event and the rows with the outcomes of the second event. The boxes are filled with the result of the calculation of the corresponding headers.
Sample space diagrams can be used when a calculation is performed with the two events. An example of this would be if we spun two spinners with numerical values on them and added the result of each outcome. Sample space diagrams are excellent for calculating probabilities of events as the number of outcomes is calculated by:
counting the number of squares containing the desired outcome.
multiplying the number of rows by the number of columns.
dividing the first number by the second number.
Two six-sided dice are rolled and the numbers obtained from each dice roll are added together. Display all of the possible outcomes with a sample space diagram.
Solution:
The result of each dice is a number from 1 to 6. We will list each of these outcomes in a table:
Each dice roll is added together, so we add up the column and row headings like this:
| 1 | 2 | 3 | 4 | 5 | 6 |
1 | 1 + 1 = 2 | 2 + 1 = 3 | 4 | 5 | 6 | 7 |
2 | 1 + 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 |
Systematic Listing of Outcomes Examples
When should a systematic listing of outcomes be utilised? When an event is described that has a large number of outcomes or permutations, a systematic listing of outcomes should be used to list all of the possible outcomes. Systematic listing of outcomes is also useful when finding probabilities of certain outcomes. We will look at some examples of situations where the systematic listing of outcomes is appropriate.
Two three-sided spinners containing the numbers 1, 2, and 3 are rolled and the result of each spin is recorded, forming a 2-digit number. What are the possible numbers that can be made?
Solution:
In this situation, there are 2 digits in the final number and each digit has 6 different possible values, meaning that there are possible numbers that can be made. This is a large number of outcomes so a systematic listing of the outcomes should be used.
We should start by making the first digit as 1, then listing all the possible outcomes like this:
11
12
13
Next, we make the first digit equal to 2 and list the possible outcomes:
21
22
23
Now we repeat this process, by having 3 as the first digit:
31
32
33
Two six-sided dice are rolled and the results of each dice roll are added together. What is the probability that the dice rolls add to 7?
Solution:
In this situation, we have two events that are being combined to form an outcome, by adding them together. This means that a sample space diagram is perfect here as they are excellent for finding probabilities of outcomes.
Begin by creating a table with headings listing the outcomes of each dice roll:
Next, fill each box with the sum of its respective column and row heading:
| 1 | 2 | 3 | 4 | 5 | 6 |
1 | 1 + 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 |
In order to find the probability of the result being 7, simply count the number of boxes that contain the number 7, then divide by the total number of boxes, which is the number of rows multiplied by the number of columns.
The probability of the result being 7 is
Importance of Systematic Listing of Outcomes
Why do we use the systematic listing of outcomes? If we simply pick outcomes at random or without method, it is likely that some outcomes may be missed out initially meaning a lot of time is spent listing them, or they may even be missed out completely. Systematic listing of outcomes makes the process of listing the outcome of events as accurate and efficient as possible. The more outcomes there are, the more effective a systematic method of listing becomes. If you wanted to see for yourself the importance of the systematic listing of outcomes, try one of the example questions from this article without using a systematic method and compare how long it takes to list all of the outcomes to a systematic method.
Systematic Listing of Outcomes - Key takeaways
- 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 is used when an outcome is made up of a combination of events that result in a large number of possible outcomes.
- Systematic listing of outcomes makes the process of listing the outcome of events as accurate and efficient as possible.
- 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 created by creating a table, heading the columns with the outcomes of the first event and the rows with the outcomes of the second event. The boxes are filled with the result of the calculation of the corresponding headers.
- Sample space diagrams can be used to calculate probabilities of outcomes by doing the following: counting the number of squares containing the desired outcome, multiplying the number of rows by the number of columns then dividing the first number by the second number.
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