Lesson 58: Probability and Chance

New Concept

Probability is the measure of how likely an event is to happen. It is expressed as a number between 0 (impossible) and 1 (certain).

If all outcomes of an experiment or game have the same probability, then the probability of an event is:

What’s next

This is just the foundation. Soon, you'll apply this formula to worked examples involving spinners, number cubes, and complementary events to master this concept.

Property

Probability is a number from 0 to 1 that measures the likelihood of an event. An event that is certain to occur has a probability of 1. An event that is certain not to occur has a probability of 0.

Examples

$P(\text{rolling a 7 on a 6-sided die}) = 0$
$P(\text{rolling a number less than 7 on a 6-sided die}) = 1$
$P(\text{flipping a coin and getting tails}) = \frac{1}{2}$

Explanation

Think of probability as a 'likeliness meter' for uncertain events. Is it definitely going to snow tomorrow (a probability of 1)? No way (a probability of 0)? Or is it a maybe, like a 50/50 coin flip? The closer the number is to 1, the more likely something is to happen. It's the mathematical way of saying 'probably'!

Property

The set of possible outcomes for an event is called the sample space.

Examples

Rolling a standard number cube: Sample space = \{1, 2, 3, 4, 5, 6\}

Explanation

Before you can find the probability of anything, you need to know all the things that could happen. The sample space is just a fancy list of every single possible result. Are you flipping a coin? The sample space is just {Heads, Tails}. It's your map of the entire 'world' of possibilities for that one experiment.

Property

If all outcomes of an experiment have the same probability, then the probability of an event is:

Examples

Probability of rolling an odd number on a die: $P(\text{odd}) = \frac{\text{outcomes } \{1, 3, 5\}}{6 \text{ total outcomes}} = \frac{3}{6} = \frac{1}{2}$
Probability of drawing a King from a 52-card deck: $P(\text{King}) = \frac{4}{52} = \frac{1}{13}$
From a bag with 5 red and 7 other marbles, the probability of drawing red is: $P(\text{red}) = \frac{5}{12}$

Explanation

When every outcome has a fair shot, calculating probability is just a simple fraction! Count how many ways you can 'win' (the outcomes you want) and put that number on top. Then, count all the things that could possibly happen (the whole sample space) and put that on the bottom. Voilà, you’ve got the probability of your event.