2 Basic Probability Concepts
In this chapter, we’ll introduce some fundamental concepts in probability theory. By the end of this chapter, you will be able to define the following:
- Sample space
- Outcome
- Event
- Probability
- Random variable
2.1 What is probability?
What is probability?
Probability is a mathematical framework for quantifying uncertainty. It assigns numerical values between 0 and 1 to events, where 0 indicates impossibility and 1 indicates certainty.
The sample space is the set of all possible outcomes of an experiment. For a coin flip, the sample space is \(\{H, T\}\).
Let’s consider a simple example. Imagine you’re flipping a fair coin. The sample space consists of two possible outcomes: heads (H) and tails (T).
The probability of getting heads is:
\[P(H) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{1}{2} = 0.5\]
2.2 Random Variables
A random variable is a function that assigns numerical values to the outcomes of a random experiment.
Can you give an example of a random variable?
Consider rolling a six-sided die. Let \(X\) be the random variable representing the number that appears on the top face. Then \(X\) can take values \(\{1, 2, 3, 4, 5, 6\}\), each with probability \(1/6\) if the die is fair.
2.3 Embedding Videos
Here’s how to embed your Manim animations. Replace your_video.mp4 with the actual filename.