Counting principle

Learning Objectives Define and identify the components of the Fundamental Counting Principle. Apply the Fundamental Counting Principle to determine the total number of possible outcomes for a sequence of independent events. Construct tree diagrams to visually represent and count outcomes for simple scenarios. Solve real-world problems involving multiple choices using the Counting Principle. Distinguish between situations where choices are independent and where they might be dependent (without explicitly using permutation/combination terms). Avoid common errors such as adding instead of multiplying when applying the Counting Principle. Ever wonder how many different outfits you can make with just a few shirts and pants? 👕👖 Or how many unique phone numbers are possible? 📱 L...

TermDefinitionExample OutcomeA single possible result of an experiment or situation.If you flip a coin, 'Heads' is an outcome. If you roll a die, '3' is an outcome. EventA specific action or occurrence that has a set of possible outcomes.Flipping a coin is an event. Choosing a shirt from your closet is an event. Sample SpaceThe set of all possible outcomes for an event or a series of events.For flipping a coin, the sample space is {Heads, Tails}. For rolling a standard die, the sample space is {1, 2, 3, 4, 5, 6}. Independent EventsEvents where the outcome of one event does not affect the outcome of another event.Flipping a coin and then rolling a die are independent events; the coin flip doesn't change the die roll possibilities. Fundamental Counting PrincipleA ru...

Fundamental Counting Principle (Two Events) $$N = n_1 \times n_2$$ If there are $n_1$ ways for the first event to occur and $n_2$ ways for the second event to occur, then there are $N$ total ways for both events to occur in sequence. This rule applies when the events are independent. Fundamental Counting Principle (Multiple Events) $$N = n_1 \times n_2 \times n_3 \times \dots \times n_k$$ If there are $n_1$ ways for the first event, $n_2$ ways for the second, and so on, up to $n_k$ ways for the $k$-th event, then the total number of ways for all $k$ independent events to occur is the product of the number of ways for each event.