Probability Rules

Once we understand what probability is, the next step is learning how to combine and manipulate probabilities.

Probability rules allow us to answer questions like:

• What is the chance of this OR that?
• What is the chance of this AND that?
• What is the chance that something does NOT happen?

These rules are essential for school exams, competitive exams, real-world decision making, statistics, and machine learning.

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Why Probability Rules Are Important

Real-world problems rarely involve just one simple event.

Probability rules help us:

• Combine multiple events correctly
• Avoid double counting
• Handle complex situations logically
• Build advanced probability models

Without rules, probability quickly becomes misleading.

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Basic Probability Properties

Every probability must satisfy:

• 0 ≤ P(A) ≤ 1
• P(S) = 1 (sample space)
• P(∅) = 0 (impossible event)

These properties always hold.

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Complement Rule

The complement of an event A means “A does NOT happen”.

It is written as:

A′ or A^c

The complement rule states:

P(A′) = 1 − P(A)

This is one of the most useful rules.

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Example: Complement Rule

If the probability of rain today is 0.3, then the probability that it does NOT rain is:

1 − 0.3 = 0.7

This idea appears frequently in exams.

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Addition Rule (OR Rule)

The addition rule is used when we want:

Probability of A OR B

This means at least one of the events occurs.

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Addition Rule (General Form)

For any two events A and B:

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

The subtraction avoids double counting.

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Why Do We Subtract the Intersection?

When adding P(A) and P(B), the outcomes common to both events get counted twice.

Subtracting P(A ∩ B) fixes this mistake.

This logic is critical for accuracy.

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Example: Addition Rule

Suppose:

• P(A) = 0.4
• P(B) = 0.5
• P(A ∩ B) = 0.2

Then:

P(A ∪ B) = 0.4 + 0.5 − 0.2 = 0.7

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Mutually Exclusive Events

Two events are mutually exclusive if they cannot occur at the same time.

That means:

P(A ∩ B) = 0

Examples:

• Getting Head or Tail in a single coin toss
• Rolling a 2 or a 5 in one die roll

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Addition Rule for Mutually Exclusive Events

If A and B are mutually exclusive:

P(A ∪ B) = P(A) + P(B)

This is a simplified version of the OR rule.

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Multiplication Rule (AND Rule)

The multiplication rule is used when we want:

Probability of A AND B

This means both events occur.

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Independent Events

Two events are independent if the occurrence of one does not affect the other.

For independent events:

P(A ∩ B) = P(A) × P(B)

This rule is very important.

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Example: Independent Events

Toss a coin and roll a die.

• P(Head) = 1/2
• P(4 on die) = 1/6

Since they are independent:

P(Head AND 4) = (1/2) × (1/6) = 1/12

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Dependent Events

Events are dependent if one event affects the probability of another.

This happens often in real life.

Example:

• Drawing cards without replacement

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Conditional Probability

Conditional probability measures the probability of an event given that another event has already occurred.

It is written as:

P(A | B)

Read as “Probability of A given B”.

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Conditional Probability Formula

The formula is:

P(A | B) = P(A ∩ B) / P(B)

This formula connects AND and conditional probability.

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Example: Conditional Probability

Suppose:

• P(A ∩ B) = 0.2
• P(B) = 0.5

Then:

P(A | B) = 0.2 / 0.5 = 0.4

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Relationship Between Independence and Conditional Probability

If A and B are independent:

P(A | B) = P(A)

This means knowing B gives no extra information about A.

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Probability Rules in Daily Life

Probability rules are used when:

• Assessing insurance risk
• Predicting weather
• Evaluating medical tests
• Making business decisions

Good decisions rely on correct rules.

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Probability Rules in Competitive Exams

Exams commonly test:

• OR rule vs AND rule
• Independent vs dependent events
• Complement and conditional probability

Careful reading of the question is crucial.

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Probability Rules in Data Science

Data science uses probability rules to:

• Combine uncertainties
• Model real-world randomness
• Estimate likelihoods

All statistical models rely on these rules.

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Probability Rules in Machine Learning

Machine learning models:

• Combine probabilities
• Use conditional probabilities
• Estimate likelihoods of outcomes

Bayesian models depend heavily on these rules.

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Common Mistakes to Avoid

Students often make these mistakes:

• Using addition instead of multiplication
• Forgetting to subtract intersection
• Confusing independence with mutual exclusivity

Always identify the event type first.

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Practice Questions

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Quick Quiz

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Quick Recap

• Complement rule: 1 − P(A)
• Addition rule handles OR events
• Multiplication rule handles AND events
• Conditional probability handles dependence
• Foundation for statistics and AI

With probability rules mastered, you are now ready to explore Combinatorics, where counting meets probability.