Basic Probability Rules

Why This Matters

Probability rules form the backbone of statistical reasoning—and you'll see them everywhere on your exam. Whether you're calculating the likelihood of combined events, updating predictions with new information, or working backward from outcomes to causes, these rules give you the mathematical toolkit to handle uncertainty. The concepts here connect directly to hypothesis testing, expected value calculations, and statistical inference, so mastering them now pays dividends throughout the course.

Here's the key insight: probability rules aren't random formulas to memorize. They're logical tools designed for specific situations. You're being tested on your ability to recognize which rule applies and why it works in a given scenario. Don't just memorize formulas—know what type of relationship between events each rule addresses and when to reach for it.

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Combining Events: The Addition Rules

When you need to find the probability that at least one of several events occurs, you're looking at a union of events. The key question is whether those events can happen simultaneously—this determines which formula you need.

Addition Rule for Mutually Exclusive Events

• Mutually exclusive events cannot occur together—if one happens, the other is impossible (like rolling a 3 or a 5 on a single die)
• Formula: $P(A \text{ or } B) = P(A) + P(B)$—no overlap means no double-counting
• Recognition cue: Look for phrases like "cannot both occur" or scenarios where outcomes are physically distinct

Addition Rule for Non-Mutually Exclusive Events

• Overlapping events require subtraction—when A and B can both occur, adding their probabilities counts the overlap twice
• Formula: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$—the subtraction corrects for double-counting
• Classic example: Drawing a red card or a face card from a deck—red face cards get counted in both categories

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Joint Occurrence: The Multiplication Rules

When you need the probability that both events occur, you're calculating an intersection. The critical distinction here is whether knowing one event occurred changes the probability of the other.

Multiplication Rule for Independent Events

• Independent events don't influence each other—the outcome of one provides no information about the other
• Formula: $P(A \text{ and } B) = P(A) \times P(B)$—simply multiply the individual probabilities
• Test for independence: Check whether $P(B|A) = P(B)$; if true, the events are independent

Multiplication Rule for Dependent Events

• Dependent events require conditional probability—the first outcome changes the probability landscape for the second
• Formula: $P(A \text{ and } B) = P(A) \times P(B|A)$, where $P(B|A)$ is the probability of B given A has occurred
• Classic scenario: Drawing cards without replacement—removing one card changes the deck composition

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Working with Conditions: Conditional Probability and Bayes' Theorem

These rules handle situations where you have partial information. Conditional probability asks "what's the probability given what I already know?" while Bayes' theorem lets you reverse the conditioning direction.

Conditional Probability

• Restricts the sample space—you're only considering outcomes where the given event has occurred
• Formula: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$—the intersection divided by the condition
• Interpretation: Think of it as zooming in on the "B happened" universe and finding A's proportion within it

Bayes' Theorem

• Reverses conditional probability—lets you find $P(A|B)$ when you know $P(B|A)$ instead
• Formula: $P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}$—updates your prior belief $P(A)$ with new evidence
• High-value applications: Medical testing, spam filters, any scenario asking "given this result, what caused it?"

Law of Total Probability

• Breaks complex probabilities into cases—partitions the sample space into mutually exclusive scenarios
• Formula: $P(A) = \sum P(A|B_i) \times P(B_i)$, where the $B_i$ events cover all possibilities
• Often paired with Bayes: Use this to calculate the denominator $P(B)$ when it's not given directly

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Simplifying Strategies: Complements and Set Operations

Sometimes the easiest path to an answer is indirect. The complement rule and set notation give you alternative approaches that can dramatically simplify calculations.

Complement Rule

• "At least one" problems become simple—instead of adding many cases, subtract the "none" case from 1
• Formula: $P(A') = 1 - P(A)$, or equivalently, $P(A) = 1 - P(A')$
• Power move: For $P(\text{at least one success in } n \text{ trials})$, calculate $1 - P(\text{all failures})$

Probability of Union of Events

• Union means "or"—at least one of the events occurs
• Formula: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$—the general addition rule in set notation
• Extends to multiple events: More events means more inclusion-exclusion corrections

Probability of Intersection of Events

• Intersection means "and"—both events occur simultaneously
• Formula: $P(A \cap B) = P(A) \times P(B|A)$ for dependent events, or $P(A) \times P(B)$ for independent
• Venn diagram connection: The intersection is the overlapping region where both circles meet

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Quick Reference Table

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Self-Check Questions

1. You're told $P(A) = 0.4$, $P(B) = 0.3$, and $P(A \text{ and } B) = 0.12$. Are A and B independent? Which multiplication rule applies, and how do you know?

2. Compare the addition rule for mutually exclusive events with the general addition rule. When does the simpler version work, and what error would you make using it incorrectly?

3. A medical test has a 95% detection rate for a disease that affects 1% of the population. If someone tests positive, what rule would you use to find the probability they actually have the disease? What other information do you need?

4. You want to find the probability of getting at least one head in five coin flips. Explain why the complement rule is more efficient than direct calculation, and set up the solution.

5. Given a two-way table showing gender and major preferences, how would you calculate $P(\text{female}|\text{STEM major})$ versus $P(\text{STEM major}|\text{female})$? What's the relationship between these two values?