How Random Is Random? Understanding Probability and Events

Ever flipped a coin and wondered: Shouldn’t it be 50/50? Then why did I get 3 heads in a row?

Welcome to the world of randomness — where short-term surprises often give way to long-term predictability.

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📚 This post is part of the "Intro to Statistics" series

🔙 Previously: Regression: Predicting Relationships Between Variables with the Best Fit Line

🔜 Next: Making Sense of Probabilities: Union, Tables, and Conditional Thinking

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🎲 What is Randomness?

Randomness means outcomes vary in unpredictable ways — in the short run.

But here’s the twist:

✅ The more you repeat a random process, the more predictable the overall pattern becomes.

Example:

• Flip a coin 5 times → may get 3 heads, 2 tails
• Flip it 5,000 times → you’ll get close to 50% heads

📊 Visual: Randomness in Small vs Large Samples

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🔍 To Measure Randomness, We Use Probability

Probability is a number between 0 and 1 that describes the likelihood of an event.

\[ 0 < P(\text{event}) < 1 \]

• 0 → impossible
• 1 → certain
• 0.5 → equal chance

For any complete experiment:

\[ \sum P(\text{all possible events}) = 1 \]

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📈 Relative Frequency & Cumulative Proportion

When observing events over time, we can estimate probability with relative frequency:

\[ P(\text{event}) = \frac{\text{event count}}{\text{total trials}} \]

The cumulative proportion is the running total of relative frequencies as trials increase — a clearer picture of probability.

🧠 Over time, the cumulative proportion settles down and reflects the true likelihood.

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🧪 Basic Terms You Must Know

• Experiment: A repeatable process with uncertain outcome
  e.g. rolling a die

• Event: A specific result or set of results
  e.g. rolling a 6, or getting an even number

• Independent Events: The result of one doesn’t affect the other
  e.g. two separate coin tosses

• Sample Space: The set of all possible outcomes
  e.g. {1, 2, 3, 4, 5, 6} for a die

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🌳 Tree Diagram Example

Let’s say you toss a coin and then roll a die.

       Start
        |
    -----------
   H           T
 / | \       / | \
1  2  3     1  2  3 ...

Each branch represents a possible compound outcome (e.g. H-2), and we can multiply probabilities along the paths.

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🧩 Types of Events

Mutually Exclusive

Events that cannot happen at the same time
Example: Rolling a 2 and a 5 in one roll

\[ P(A \cap B) = 0 \]

Collectively Exhaustive

Events that together cover all possibilities
\[ \sum P(\text{events}) = 1 \]

Complement Event

All outcomes not in the event
\[ P(A^c) = 1 - P(A) \]

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🔗 Disjoint vs Independent

Type	Description	Rule
Disjoint (Mutually Exclusive)	Events don’t overlap	\( P(A \cap B) = 0 \)
Independent	Events don’t affect each other	\( P(A \cap B) = P(A) \cdot P(B) \)

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🔍 Venn Diagram: Disjoint vs Overlapping

• Disjoint events → no overlap
• Independent events → can overlap, but still multiply their probabilities

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🧠 Level Up: Why This Matters

Understanding randomness and events helps you:

• 🎯 Build simulations for decision-making
• 🧠 Model uncertainty in machine learning and forecasting
• 🔐 Analyze risk in finance, health, or engineering

Next, we’ll look at probability rules — like the addition and multiplication rule — and how they apply in real problems.

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📌 Try It Yourself: Randomness & Probability

Q1: What does randomness mean in statistics?

💡 Show Answer

It refers to unpredictable outcomes in the short term, but stable patterns in the long run.

Q2: What is the range of any valid probability value?

💡 Show Answer

Between 0 and 1 (i.e., 0 ≤ P(event) ≤ 1).

Q3: What is a sample space?

💡 Show Answer

The set of all possible outcomes for an experiment.

Q4: If two events are mutually exclusive, what does that imply?

💡 Show Answer

They cannot both occur at the same time. So, P(A ∩ B) = 0.

Q5: What is the formula for relative frequency?

💡 Show Answer

P(event) = event count ÷ total number of trials.

Q6: What is the probability of two independent events A and B both occurring?

💡 Show Answer

P(A ∩ B) = P(A) × P(B) — because they don’t affect each other.

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🧠 Summary

Concept	Meaning
Randomness	Unpredictable short-term, predictable long-term
Probability	Likelihood of an event (between 0 and 1)
Relative Frequency	Estimate based on observed trials
Sample Space	All possible outcomes
Mutually Exclusive	Can’t happen together
Collectively Exhaustive	Together cover all possibilities
Complement	All other outcomes
Independent	One event doesn’t affect the other
Tree Diagram	Shows compound outcomes
Venn Diagram	Visualize event relationships

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✅ Up Next

In the next post, we’ll explore:

• Addition Rule for combined events
• Multiplication Rule for independent events
• Conditional Probability and its meaning

Stay curious!