Probability - Seventh Grade

Simple & Compound Events, Theoretical & Experimental

1. Sample Spaces and Simple Events

What is a Sample Space?

Sample Space is the set of

ALL POSSIBLE OUTCOMES

• Usually denoted by S

• Lists every outcome that could happen

Examples of Sample Spaces

Flipping a coin: S = {Heads, Tails}

Rolling a die: S = {1, 2, 3, 4, 5, 6}

Spinner with 4 colors: S = {Red, Blue, Green, Yellow}

What is an Event?

Simple Event: A single outcome from the sample space

Example: Rolling a 5 on a die

Compound Event: Two or more simple events combined

Example: Rolling an even number (2, 4, or 6)

2. Probability of Simple Events

Basic Probability Formula

P(Event) = Favorable Outcomes / Total Outcomes

P(E) = n(E) / n(S)

Important Properties:

• Probability is always between 0 and 1

• 0 ≤ P(E) ≤ 1

• P = 0 means IMPOSSIBLE event

• P = 1 means CERTAIN event

• Can be written as fraction, decimal, or percent

Example

Problem: What is the probability of rolling a 4 on a standard die?

Favorable outcomes: 1 (only the number 4)

Total outcomes: 6 (numbers 1, 2, 3, 4, 5, 6)

P(rolling 4) = 1/6 ≈ 0.167 or 16.7%

Answer: P = 1/6

3. Opposite Events (Complementary Events)

What are Opposite Events?

Opposite (Complementary) Events are

ALL outcomes that are NOT the event

• Denoted as P(not E) or P(E')

• Event and its complement cover ALL possibilities

Complementary Probability Formula

P(not E) = 1 - P(E)

P(E) + P(not E) = 1

Example

Problem: If P(rain) = 0.3, what is P(no rain)?

P(no rain) = 1 - P(rain)

P(no rain) = 1 - 0.3

P(no rain) = 0.7 or 70%

Answer: P(no rain) = 0.7

4. Mutually Exclusive vs. Overlapping Events

Mutually Exclusive Events

Events that CANNOT happen at the same time

Example: Rolling a 3 AND rolling a 5 (can't happen together)

For Mutually Exclusive Events:

P(A or B) = P(A) + P(B)

Overlapping Events

Events that CAN happen at the same time

Example: Drawing a red card AND drawing a king (red king exists)

For Overlapping Events:

P(A or B) = P(A) + P(B) - P(A and B)

Subtract overlap to avoid counting twice!

Example

Problem: Probability of rolling less than 3 OR rolling an even number on a die.

Less than 3: {1, 2} → P(A) = 2/6

Even: {2, 4, 6} → P(B) = 3/6

Both (overlap): {2} → P(A and B) = 1/6

P(A or B) = 2/6 + 3/6 - 1/6 = 4/6 = 2/3

Answer: P = 2/3

5. Theoretical vs. Experimental Probability

Theoretical Probability

Based on WHAT SHOULD HAPPEN

P = Favorable / Total Possible

Uses mathematical reasoning

Experimental Probability

Based on WHAT ACTUALLY HAPPENED

P = Times Event Occurred / Total Trials

Uses actual data from experiments

Feature	Theoretical	Experimental
Based on	Math/Logic	Real data
When to use	Before experiment	After experiment
Accuracy	Perfect (ideal)	Varies by trials

Example

Flipping a coin:

Theoretical: P(heads) = 1/2 = 0.5

Experimental: Flipped 50 times, got 28 heads

P(heads) = 28/50 = 0.56

Not exactly the same, but close! More trials → closer to theoretical

6. Compound Events and Sample Spaces

What are Compound Events?

Compound events involve

TWO OR MORE simple events happening together

Example: Flipping a coin AND rolling a die

Counting Outcomes: Fundamental Counting Principle

Total Outcomes = n₁ × n₂ × n₃...

Multiply the number of outcomes for each event

Methods to List Sample Space

1. Organized List: Write all possibilities

2. Tree Diagram: Branch out each choice

3. Table/Grid: Create rows and columns

Example

Problem: Flip a coin and roll a die. How many outcomes?

Coin outcomes: 2 (H, T)

Die outcomes: 6 (1, 2, 3, 4, 5, 6)

Total = 2 × 6 = 12 outcomes

Sample Space: {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

Answer: 12 outcomes

7. Independent and Dependent Events

Independent Events

Events where one does NOT affect the other

• Outcome of first event doesn't change probability of second

Example: Flipping a coin twice (first flip doesn't affect second)

For Independent Events:

P(A and B) = P(A) × P(B)

Multiply the probabilities

Dependent Events

Events where one DOES affect the other

• Outcome of first event changes probability of second

Example: Drawing cards without replacement

For Dependent Events:

P(A and B) = P(A) × P(B after A)

Second probability changes based on first outcome

Example: Independent

Problem: Flip a coin twice. What's P(heads then tails)?

P(heads) = 1/2

P(tails) = 1/2

P(H then T) = 1/2 × 1/2 = 1/4

Answer: P = 1/4 or 0.25

Example: Dependent

Problem: Bag has 3 red and 2 blue marbles. Draw 2 without replacement. What's P(both red)?

First draw: P(red) = 3/5

Second draw: Now only 2 red left out of 4 total

P(red after red) = 2/4 = 1/2

P(both red) = 3/5 × 1/2 = 3/10

Answer: P = 3/10 or 0.3

8. Making Predictions

Prediction Formula

Expected Outcome = Probability × Number of Trials

Use this to predict:

How many times an event will occur in a certain number of trials

Example

Problem: If you roll a die 60 times, about how many times will you roll a 3?

P(rolling 3) = 1/6

Number of trials = 60

Expected = 1/6 × 60 = 10

Answer: About 10 times

Quick Reference: All Probability Formulas

Type	Formula
Basic Probability	P(E) = Favorable / Total
Complementary	P(not E) = 1 - P(E)
Mutually Exclusive (OR)	P(A or B) = P(A) + P(B)
Overlapping (OR)	P(A or B) = P(A) + P(B) - P(A and B)
Independent (AND)	P(A and B) = P(A) × P(B)
Dependent (AND)	P(A and B) = P(A) × P(B after A)
Counting Outcomes	n₁ × n₂ × n₃...
Experimental	Times Occurred / Total Trials
Prediction	P(E) × Number of Trials