Lesson 32: Probability

New Concept

Probability measures the chance of something happening! It is the likelihood that a particular event will occur and is calculated as a ratio of favorable outcomes to possible outcomes:

What’s next

You'll apply this formula to calculate odds and map out all possible outcomes (the sample space). Soon, we'll compare theoretical predictions to real-world experimental results.

Property

Probability is the likelihood that a particular event will occur. We express probability as a number ranging from zero to one.

Examples

• Rolling an even number on a 6-sided die, the probability is: $P(\text{even}) = \frac{3}{6} = \frac{1}{2}$
• Picking a blue marble from a bag with 5 red and 3 blue ones: $P(\text{blue}) = \frac{3}{8}$

Explanation

Think of probability as a score for how likely something is, from 0 (no way!) to 1 (for sure!). It’s a fraction comparing what you want to happen to everything that could happen. This helps you predict outcomes in games or experiments!

Property

The sample space of an experiment is the collection of all possible outcomes. We can record the sample space in a variety of ways, including a list or a table.

Examples

• For flipping a coin twice, the sample space is: \{HH, HT, TH, TT\}
• For rolling a standard six-sided die, the sample space is: \{1, 2, 3, 4, 5, 6\}

Explanation

The sample space is your ultimate cheat sheet for any probability problem. It’s a complete list of every single possible result! Before you can figure out the chances of something happening, you need to know all the things that could happen. Listing them all makes calculating probabilities much easier.

Property

The complement of an event is the set of outcomes in the sample space that are not included in the event.

Examples

• If the chance of rain is $0.2$, the chance of no rain is: $1 - 0.2 = 0.8$
• The probability of rolling a 6 is $\frac{1}{6}$. The probability of not rolling a 6 is: $1 - \frac{1}{6} = \frac{5}{6}$

Explanation

The complement is a super useful shortcut for finding the probability of something not happening. Since an event either occurs or it doesn't, their probabilities always add up to 1. Just subtract the event's probability from 1 to find its opposite chance!