Counting principle

Learning Objectives Define the Fundamental Counting Principle and distinguish it from the Addition Principle. Apply the Multiplication Principle to calculate the total number of outcomes for a sequence of independent events. Solve multi-stage counting problems that include specific restrictions or conditions. Use the counting principle to determine the size of a sample space in a probability problem. Construct tree diagrams to visualize and verify the outcomes of simple multi-stage experiments. Differentiate between problems requiring the Addition Principle ('or') and those requiring the Multiplication Principle ('and'). How many different 4-digit phone passcodes are possible? 🤔 Listing them all would be impossible, but the counting principle gives us th...

TermDefinitionExample OutcomeA single possible result of a probability experiment.When rolling a standard six-sided die, rolling a '4' is one possible outcome. Sample SpaceThe set of all possible outcomes of a probability experiment.The sample space for flipping a coin is {Heads, Tails}. Independent EventsTwo or more events where the outcome of one event does not affect the outcome of the other(s).Flipping a coin and rolling a die are independent events. The coin's result doesn't change the die's possible outcomes. Tree DiagramA diagram used to visualize and list all possible outcomes of a sequence of events. Each branch represents a possible outcome.To show the outcomes of flipping a coin twice, the first level has two branches (H, T), and each of those branches...

The Multiplication Principle (Fundamental Counting Principle) If an event A can occur in `m` ways and an independent event B can occur in `n` ways, then the total number of ways that both A and B can occur is `m \times n`. Use this rule when a process involves a sequence of choices or stages, and you need to find the total number of outcomes for the entire sequence (an 'and' situation). The Addition Principle If an event A can occur in `m` ways and an event B can occur in `n` ways, and A and B are mutually exclusive (cannot happen at the same time), then the number of ways that A or B can occur is `m + n`. Use this rule when you are making a single choice from two or more distinct, non-overlapping groups (an 'or' situation).