Basic Probability Rules

Addition and multiplication rules

Basic Probability Rules

Probability Basics

Probability: Measure of likelihood an event occurs (0 to 1)

P(A) = 0 → Event A is impossible
P(A) = 1 → Event A is certain
0 < P(A) < 1 → Event A may or may not occur

Complement: Event A doesn't occur, denoted $A^c$ or $\bar{A}$

$P(A^c) = 1 - P(A)$

Sample Space and Events

Sample Space (S): Set of all possible outcomes
Event: Subset of sample space

Example: Roll a die

Sample space: S = {1, 2, 3, 4, 5, 6}
Event "even number": E = {2, 4, 6}
P(E) = 3/6 = 0.5

Computing Probability

Equally likely outcomes:

$P(A) = \frac{\text{Number of outcomes in A}}{\text{Total number of outcomes}}$

Example: Deck of cards, P(Heart) = 13/52 = 1/4

Relative frequency (empirical probability):

$P(A) \approx \frac{\text{Number of times A occurred}}{\text{Total number of trials}}$

Addition Rule (OR)

For any two events A and B:

$P(A \text{ or } B) = P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Why subtract P(A ∩ B)? Avoid double-counting outcomes in both A and B

Example: Draw one card

P(Heart) = 13/52
P(Face card) = 12/52
P(Heart and Face) = 3/52
P(Heart or Face) = 13/52 + 12/52 - 3/52 = 22/52

Mutually Exclusive Events

Definition: Events that cannot both occur (no overlap)

If A and B are mutually exclusive: P(A ∩ B) = 0

Addition Rule simplifies:

$P(A \text{ or } B) = P(A) + P(B)$

Example: Roll a die

Event A: Roll 2
Event B: Roll 5
These are mutually exclusive (can't roll both)
P(A or B) = 1/6 + 1/6 = 2/6 = 1/3

Non-example: P(Heart) and P(Ace) are NOT mutually exclusive (Ace of Hearts is in both)

Multiplication Rule (AND)

For any two events:

$P(A \text{ and } B) = P(A \cap B) = P(A) \times P(B|A)$

Where P(B|A) = probability of B given A occurred

We'll explore this more in conditional probability topic

Independent Events (Preview)

If A and B are independent: P(B|A) = P(B)

Multiplication Rule simplifies:

$P(A \text{ and } B) = P(A) \times P(B)$

Example: Flip coin twice

P(First heads) = 1/2
P(Second heads) = 1/2
P(Both heads) = 1/2 × 1/2 = 1/4

Probability Rules Summary

Rule 1: For any event A, 0 ≤ P(A) ≤ 1

Rule 2: P(S) = 1 (something in sample space must occur)

Rule 3: Complement Rule: P(A^c) = 1 - P(A)

Rule 4: Addition Rule: P(A or B) = P(A) + P(B) - P(A and B)

Rule 5: If mutually exclusive: P(A or B) = P(A) + P(B)

Rule 6: Multiplication Rule: P(A and B) = P(A) × P(B|A)

Rule 7: If independent: P(A and B) = P(A) × P(B)

Venn Diagrams

Visual tool for probability:

Union (A or B): Everything in A or B or both
Intersection (A and B): Overlap of A and B
Complement (A^c): Everything outside A

Use Venn diagrams to visualize and organize probability problems.

Tree Diagrams

Useful for sequential events:

Each branch represents outcome
Multiply probabilities along path
Add probabilities of different paths to same outcome

Example: Flip coin twice

First flip: 1/2 Heads, 1/2 Tails
Second flip: 1/2 Heads, 1/2 Tails
P(HH) = 1/2 × 1/2 = 1/4
P(exactly one head) = P(HT) + P(TH) = 1/4 + 1/4 = 1/2

Common Mistakes

❌ Adding probabilities when should multiply (AND vs OR confusion)
❌ Forgetting to subtract overlap in addition rule
❌ Assuming events are independent when they're not
❌ Confusing mutually exclusive with independent

Practice Strategy

Identify: What event(s) are we finding probability for?
Determine: AND (multiply) or OR (add)?
Check: Mutually exclusive? Independent?
Calculate: Apply appropriate rule
Verify: Does answer make sense (between 0 and 1)?

Quick Reference

Complement: P(A^c) = 1 - P(A)
OR (Addition): P(A or B) = P(A) + P(B) - P(A and B)
AND (Multiplication): P(A and B) = P(A) × P(B|A)
Mutually Exclusive OR: P(A or B) = P(A) + P(B)
Independent AND: P(A and B) = P(A) × P(B)

📚 Practice Problems

1Problem 1easy

❓ Question:

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of drawing: a) A red marble? b) A blue or green marble? c) Not a red marble?

💡 Show Solution

Step 1: Count total outcomes Total marbles = 5 + 3 + 2 = 10

Step 2: Calculate P(red marble) Favorable outcomes = 5 red marbles P(Red) = 5/10 = 1/2 = 0.5 or 50%

Step 3: Calculate P(blue or green) Method 1 - Addition Rule: P(Blue or Green) = P(Blue) + P(Green) = 3/10 + 2/10 = 5/10 = 1/2 = 0.5

Method 2 - Count directly: Blue or green marbles = 3 + 2 = 5 P(Blue or Green) = 5/10 = 1/2

Step 4: Calculate P(not red) Method 1 - Complement Rule: P(Not Red) = 1 - P(Red) = 1 - 5/10 = 5/10 = 1/2

Method 2 - Count directly: Not red = blue + green = 3 + 2 = 5 P(Not Red) = 5/10 = 1/2

Answer: a) P(Red) = 1/2 or 0.5 b) P(Blue or Green) = 1/2 or 0.5 c) P(Not Red) = 1/2 or 0.5

2Problem 2easy

❓ Question:

In a class of 30 students, 18 play sports and 12 play an instrument. If 5 students do both, what is the probability that a randomly selected student: a) Plays sports OR an instrument? b) Plays neither?

💡 Show Solution

Step 1: Organize the information using Venn diagram Total students = 30 Play sports (S) = 18 Play instrument (I) = 12 Both (S AND I) = 5

Step 2: Find students in each region Only sports = 18 - 5 = 13 Only instrument = 12 - 5 = 8 Both = 5 Neither = 30 - (13 + 5 + 8) = 30 - 26 = 4

Step 3: Calculate P(Sports OR Instrument) Using Addition Rule for overlapping events: P(S OR I) = P(S) + P(I) - P(S AND I) = 18/30 + 12/30 - 5/30 = (18 + 12 - 5)/30 = 25/30 = 5/6 ≈ 0.833

Or count directly: 13 + 5 + 8 = 26 P(S OR I) = 26/30 = 13/15 ≈ 0.867

Wait, let me recalculate: Only sports = 13 Both = 5
Only instrument = 8 Total in S OR I = 13 + 5 + 8 = 26

P(S OR I) = 26/30 = 13/15

Step 4: Calculate P(Neither) Method 1 - Complement: P(Neither) = 1 - P(S OR I) = 1 - 26/30 = 4/30 = 2/15 ≈ 0.133

Method 2 - Count directly: Neither = 4 students P(Neither) = 4/30 = 2/15

Answer: a) P(Sports OR Instrument) = 26/30 = 13/15 ≈ 0.867 or 86.7% b) P(Neither) = 4/30 = 2/15 ≈ 0.133 or 13.3%

3Problem 3medium

❓ Question:

Roll two fair dice. What is the probability that the sum is 7 or 11?

💡 Show Solution

Step 1: Find total possible outcomes Each die: 6 outcomes Two dice: 6 × 6 = 36 total outcomes

Step 2: Find outcomes where sum = 7 List all combinations: (1,6): 1 + 6 = 7 (2,5): 2 + 5 = 7 (3,4): 3 + 4 = 7 (4,3): 4 + 3 = 7 (5,2): 5 + 2 = 7 (6,1): 6 + 1 = 7

Count: 6 ways to get sum of 7

Step 3: Find outcomes where sum = 11 (5,6): 5 + 6 = 11 (6,5): 6 + 5 = 11

Count: 2 ways to get sum of 11

Step 4: Find P(sum is 7 OR 11) Events are mutually exclusive (can't get both 7 AND 11) Use Addition Rule: P(7 OR 11) = P(7) + P(11) = 6/36 + 2/36 = 8/36 = 2/9 ≈ 0.222

Answer: P(sum is 7 or 11) = 8/36 = 2/9 ≈ 0.222 or 22.2%

4Problem 4medium

❓ Question:

At a school, 60% of students are in band, 50% are in chorus, and 30% are in both. If a student is selected at random, what is the probability they are in band or chorus? Draw a Venn diagram.

💡 Show Solution

Step 1: Set up the information P(Band) = 0.60 P(Chorus) = 0.50 P(Band AND Chorus) = 0.30

Step 2: Use Addition Rule for overlapping events P(B OR C) = P(B) + P(C) - P(B AND C) = 0.60 + 0.50 - 0.30 = 0.80

Step 3: Verify with Venn diagram regions Only Band = P(B) - P(B AND C) = 0.60 - 0.30 = 0.30 Only Chorus = P(C) - P(B AND C) = 0.50 - 0.30 = 0.20 Both = 0.30 Neither = 1 - (0.30 + 0.20 + 0.30) = 0.20

Venn Diagram: Band (60%) Chorus (50%) ___ ___ / ___ ___/ | / | | 0.30 | 0.30 | 0.20| | / | ___/ ___/

Neither: 0.20

Step 4: Calculate P(Band OR Chorus) P(B OR C) = 0.30 + 0.30 + 0.20 = 0.80

Check: 0.30 + 0.20 + 0.30 + 0.20 = 1.00 ✓

Answer: P(Band OR Chorus) = 0.80 or 80%

Key insight: We must subtract the overlap (0.30) because students in both were counted twice when we added P(Band) + P(Chorus).

5Problem 5hard

❓ Question:

Prove that for any event A, 0 ≤ P(A) ≤ 1, and explain what P(A) = 0 and P(A) = 1 mean.

💡 Show Solution

Step 1: Prove 0 ≤ P(A) ≤ 1

Definition of probability: P(A) = (number of favorable outcomes) / (total number of outcomes)

Step 2: Prove P(A) ≥ 0 (lower bound) Number of favorable outcomes ≥ 0

Cannot have negative outcomes
Minimum is 0 (impossible event)

Total number of outcomes > 0

Sample space must have at least one outcome

Therefore: P(A) = (≥0) / (>0) ≥ 0

Step 3: Prove P(A) ≤ 1 (upper bound) Number of favorable outcomes ≤ Total outcomes

Cannot have more favorable than total
Maximum: all outcomes are favorable

Therefore: P(A) = (≤total) / (total) ≤ 1

Conclusion: 0 ≤ P(A) ≤ 1

Step 4: Interpret P(A) = 0

Meaning: Event A is IMPOSSIBLE

No favorable outcomes
Can never occur
0% chance

Examples:

P(sum = 13 with two dice) = 0
P(randomly selecting a negative number from {1,2,3,4,5}) = 0
P(flipping a coin and getting both heads and tails) = 0

Step 5: Interpret P(A) = 1

Meaning: Event A is CERTAIN

All outcomes are favorable
Must occur
100% chance

Examples:

P(sum ≤ 12 with two dice) = 1
P(selecting a positive number from {1,2,3,4,5}) = 1
P(flipping a coin and getting heads or tails) = 1

Step 6: Interpret intermediate probabilities

P(A) = 0.5: Event occurs half the time (equally likely) P(A) = 0.25: Event occurs 1/4 of the time (unlikely) P(A) = 0.75: Event occurs 3/4 of the time (likely)

The closer to 1, the more likely The closer to 0, the less likely

Step 7: Related rules

Complement rule: P(A) + P(not A) = 1

If P(A) = 0.3, then P(not A) = 0.7
Together they cover all possibilities

Sum of all probabilities = 1:

If sample space is {A, B, C}
Then P(A) + P(B) + P(C) = 1

Answer: Probability must be between 0 and 1 because it's a ratio of favorable outcomes to total outcomes. P(A) = 0 means A is impossible (never occurs). P(A) = 1 means A is certain (always occurs). Values in between represent varying degrees of likelihood.

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