Foundations of Probability

Explore fundamental concepts through simple, interactive examples.

What is Probability? 🤔

Have you ever wondered how a video game decides what item an enemy drops? For a long time, people just saw random events as pure "luck."

But game designers can't rely on luck. They need to make sure a game is fair and fun. They use probability to decide the chance of a "common" item versus a "rare" one.

For example, they might decide a monster has 100 possible items in its "drop table." To make an item "rare," they might make it just 1 of those 100 possibilities. To make an item "common," they might make it 70 of those 100 possibilities.

This simple idea—that we can *count* the "favorable" outcomes (like getting the rare item) against the "total" outcomes—is the foundation of everything we'll learn.

Probability

Probability is simply a way to measure the **chance** of something happening. Think of it as a number between 0 and 1 that tells you how likely an event is, from **impossible (0)** to **certain (1)**.

What is a Sample Space?

Sample Space

The sample space is the complete list of *all possible outcomes* of an experiment. We need to know this *first* to find a probability (like all 100 possible items a monster can drop).

For example, when you flip one coin, the sample space is simply S={Heads, Tails}S = \{\text{Heads, Tails}\}. The "Total number of possible outcomes" is 2.

Once we have the sample space, we find the probability by counting:

P(Event)=Number of favorable outcomesTotal number of possible outcomesP(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
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Core Rules of Probability

  • A probability is always between 0 and 1 (inclusive). This is because the top number (numerator) can never be larger than the bottom number (denominator).
    • P(Event)=0P(\text{Event}) = 0 means it's impossible.
    • P(Event)=1P(\text{Event}) = 1 means it's certain.
  • The probabilities of all possible outcomes must add up to 1.(e.g., P(Heads)+P(Tails)=0.5+0.5=1P(\text{Heads}) + P(\text{Tails}) = 0.5 + 0.5 = 1)
If you roll one fair six-sided die, what is the probability of rolling a 3?

Explore These Concepts! 👇

Now, let's see these ideas in action! Use the tabs below to explore different examples. You can start with a "One Coin Flip", see what happens when you roll dice, or check the chances of getting different "Party Favors" from a bag.

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Beyond Basic Counting

This "favorable / total" formula works perfectly when every outcome is **equally likely**, or when the sample space is small and we can list out all the outcomes and count (like a fair coin or a standard die).

For more complex questions, we'll learn to use powerful tools like Venn diagrams, tree diagrams, contingency tables, and random variables.

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Review Your Materials

To master these new concepts, be sure to review the corresponding class slides and complete the homework assignments!

Example: A Single Flip of a Fair Coin

1. Sample Space

The collection of all possible outcomes. For a single coin flip, where H is Heads and T is Tails, the sample space (S) is:S = {H, T}There are 2 total possible outcomes.

2. Outcomes & Events

An outcome is a single result (e.g., 'H'). An event is a set of outcomes you are interested in. For a single flip, an event and an outcome are often the same thing. For example, the event "getting a Head" is:E = {H}

3. Probability

The likelihood of an event, calculated as: P(Event) = (Number of favorable outcomes) / (Total number of outcomes).
  • Probability of getting "Heads": P(H) = 1/2 = 0.5
  • Probability of getting "Tails": P(T) = 1/2 = 0.5

Key Takeaway

The probability of a single event is the ratio of favorable outcomes to total possible outcomes. For a fair coin, P(Heads)=12P(\text{Heads}) = \frac{1}{2}.

Need help?

Try asking our AI assistant a question like: "Why is the probability of heads 0.5?"

Manual One Coin Flip Experiment

Click the button to run a single trial and see the distribution build up over time.

Click the button to start an experiment!

Run a trial to see the results and statistics.

Law of Large Numbers Simulation

This principle states that as the number of trials increases, the experimental probability will converge on the theoretical probability. Try it yourself!

Click "Run Simulation" to see the results.