Lesson 14: Determining the Theoretical Probability of an Event

New Concept

Algebra is a branch of mathematics that uses symbols to represent unknown quantities and express general relationships, turning complex problems into solvable equations.

For example, a relationship like probability is defined with a formula:

What’s next

This course will build your skills step-by-step. We'll start by using a classic algebraic formula to explore the world of theoretical probability and chance.

Property

A sample space is the set of all possible outcomes of an event.

Examples

• When tossing a fair coin, the sample space of possible outcomes is {heads, tails}.
• For a standard number cube labeled 1-6, the sample space for a single roll is {1, 2, 3, 4, 5, 6}.
• If a spinner has four equal sections (blue, yellow, green, red), the sample space is {blue, yellow, green, red}.

Explanation

Imagine listing every single possible thing that could happen in an experiment—that’s your sample space! Before you even roll a die, you know the possibilities are 1, 2, 3, 4, 5, or 6. This complete list of potential outcomes is the sample space. It's like knowing all the menu options before you order your food.

Property

Examples

• With 4 green, 3 blue, and 3 red marbles in a bag, the probability of choosing red is $\operatorname{P}(\text{red}) = \frac{3}{10} = 0.3$.
• For a number cube, the probability of rolling an odd number ({1, 3, 5}) is $\operatorname{P}(\text{odd}) = \frac{3}{6} = \frac{1}{2}$.
• In a carnival game with 16 paths, if 2 paths lead to winning 2 dollars, the probability is $\operatorname{P}(\text{2 dollars}) = \frac{2}{16} = \frac{1}{8}$.

Explanation

This is your superpower for predicting the future using math! Theoretical probability calculates the chances of something happening without you actually doing the experiment. By comparing the outcomes you want (favorable) to all the things that could happen (total outcomes), you can figure out just how likely your desired event is. It’s all about what should happen in a perfect world.

Property

A complement of an event is a set of all outcomes of an experiment that are not in a given event.

Examples

• On a number cube, the probability of rolling a 4 is $\frac{1}{6}$. The probability of not rolling a 4 is $1 - \frac{1}{6} = \frac{5}{6}$.
• In a bag with 4 green and 6 red marbles, the probability of not choosing green is $1 - \operatorname{P}(\text{green}) = 1 - \frac{4}{10} = \frac{6}{10}$.
• If the probability of rain is 25% (or 0.25), the probability that it will not rain is $1 - 0.25 = 0.75$, or 75%.

Explanation

The complement is basically 'everything else.' If you want to know the chance of something not happening, this is your go-to trick! Instead of counting all the ways an event can fail, just find the probability of it succeeding and subtract that from 1. It’s a clever shortcut for finding the probability of the opposite outcome.