Basic Counting Techniques

Section 4.1 Basic Counting Techniques

In this section we discover some fundamental counting techniques which will provide us a foundation to build upon. You’ll encounter some scenarios in which addition is utilized to count a number of objects/possibilities as well as other scenarios in which multiplication is utilized. You’ll identify how to determine which approach should be applied in a given scenario and you’ll apply your chosen method to carry out the counting.

Subsection 4.1.1 Counting Outcomes of Atomic Processes

Throughout this short subsection, we’ll use the phrase atomic processes to describe processes which cannot be decomposed into smaller, sub-processes. Examples of these atomic processes are the flip of a coin or a roll of a single die. This is as opposed to a more complex process like the rolling of a pair of dice, the flipping of multiple coins, or a combination of die rolls and coin flips. In these more complex cases, we can break the corresponding process into two or more atomic processes. For example, rolling a pair of dice consists of two atomic processes -- the roll of the first die and the roll of the second.

Verify your understanding of atomic processes versus more complex processes by answering the following embedded exercise.

Checkpoint 4.1.1. Identifying Atomic Processes.

Were you comfortable with classifying those processes? The convenient thing about what we’ve classified as atomic processes is that the number of outcomes possible from an atomic process is simple to count. Indeed, the number of outcomes possible on a single flip of a coin is two -- we can observe a heads or a tails as the outcome. Verify this by counting the number of outcomes possible for each of the atomic processes in the embedded exercise below.

Checkpoint 4.1.2. Counting Outcomes of Atomic Processes.

Good work. In the next subsection and the following section in this chapter, we’ll use what we know about counting the number of outcomes from atomic processes to help us count the number of outcomes of more complex processes with multiple components.

Subsection 4.1.2 Near-Atomic Processes: One or the Other(s)

In this subsection, we consider processes which ultimately consist of just a single action but where there is a choice or uncertainty about which action will be taken. For example, we might consider the opportunity to flip a coin or roll of a six-sided die. In this case, we either flipe the coin or roll the die, but we don’t do both. In this case, there are two possible outcomes if we flip the coin and an additional six outcomes possible if we roll the die. In total, there are eight (8) outcomes possible from this process.

We consider the process above a near-atomic process because we can’t quite break it into smaller sub-processes because only one of these sub-processes will actually be enacted. In this case, the total number of outcomes possible is the sum of the number of outcomes possible from each component process. That is, we obtained 8 total outcomes possible in the example above because there were two outcomes associated with the coin flip and six outcomes associated with the die roll. Since we were either flipping the coin or rolling the die, we added the two totals together.

Test your grasp of what constitutes a near-atomic process by completing this next interactive exercise.

Checkpoint 4.1.3. Identifying near-Atomic Processes.

Test your understanding of counting the possible number of outcomes from near-atomic processes in the interactive exercise below.

Checkpoint 4.1.4. Counting Outcomes of Atomic Processes.

As you worked through those last two exercises, make note of where you were confident as well as where you stumbled or were less sure of yourself. Be sure to ask questions on the items where you were less sure of yourself.

Good work through this introductory section on counting. See you in the section, where we’ll be introduced to the Fundamental Principle of Counting which will allow us to count the number of outcomes possible for non-atomic, complex processes where the several component processes are all completed. That is, rather than choosing to flip a coin or roll a die, we’ll flip the coin and roll the die.

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