Probability - Grade 8

1. Probability of Simple Events

Definition: The probability of a simple event is the chance that the event will occur. It's the ratio of favorable outcomes to total possible outcomes.

Formula:

\( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{n(A)}{n(S)} \)

Key Properties:

Range: \( 0 \leq P(A) \leq 1 \)
Impossible event: P(A) = 0
Certain event: P(A) = 1
Can be expressed as: Fraction, decimal, or percent

Examples:

Example 1: Rolling a 4 on a standard die

Favorable outcomes: 1 (only one 4)

Total outcomes: 6 (numbers 1-6)

\( P(\text{rolling 4}) = \frac{1}{6} \approx 0.167 \text{ or } 16.7\% \)

Example 2: Drawing a red card from a standard deck

Favorable: 26 (hearts and diamonds)

Total: 52 cards

\( P(\text{red card}) = \frac{26}{52} = \frac{1}{2} = 0.5 \text{ or } 50\% \)

2. Opposite, Mutually Exclusive, and Overlapping Events

Opposite Events (Complementary Events):

Two events are opposite if one event is "not" the other. They cover all possible outcomes.

\( P(A) + P(\text{not } A) = 1 \)

\( P(\text{not } A) = 1 - P(A) \)

Example: P(raining) and P(not raining) are opposite events

Mutually Exclusive Events:

Events that CANNOT happen at the same time. If one occurs, the other cannot.

\( P(A \text{ and } B) = 0 \)

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

Example: Rolling a 3 or a 5 on one die (can't be both)
Example: Turning left or right (can't do both)

Overlapping Events:

Events that CAN happen at the same time. They share some common outcomes.

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

Example: Drawing a card that is red OR a face card
Some cards are both (red face cards), so we subtract the overlap

Example Problem:

In a deck of 52 cards, find P(red OR face card)

P(red) = 26/52, P(face card) = 12/52, P(red AND face card) = 6/52

\( P(\text{red OR face}) = \frac{26}{52} + \frac{12}{52} - \frac{6}{52} = \frac{32}{52} = \frac{8}{13} \)

3. Experimental Probability

Definition: Probability based on actual experiments or observations, not theoretical calculations.

Formula:

\( P(\text{event}) = \frac{\text{Number of times event occurred}}{\text{Total number of trials}} \)

Comparison:

Type	Theoretical Probability	Experimental Probability
Based on	Mathematical calculation	Actual experiments/data
Formula	Favorable/Total possible	Times occurred/Total trials
Example	P(heads) = 1/2	Flipped 50 times, got 23 heads = 23/50

Example:

A basketball player shoots 80 free throws and makes 64 of them. What is the experimental probability of making a free throw?

\( P(\text{making free throw}) = \frac{64}{80} = \frac{4}{5} = 0.8 = 80\% \)

Key Point:

As the number of trials increases, experimental probability usually gets closer to theoretical probability.

4. Find Probabilities Using Two-Way Frequency Tables

Definition: A two-way frequency table shows data for two categories, organized in rows and columns.

Example Table:

Survey of 100 students about pets:

	Has Dog	No Dog	Total
Has Cat	15	20	35
No Cat	35	30	65
Total	50	50	100

Finding Probabilities:

1. P(student has a dog):

\( P(\text{dog}) = \frac{50}{100} = \frac{1}{2} = 50\% \)

2. P(student has both cat and dog):

\( P(\text{cat and dog}) = \frac{15}{100} = \frac{3}{20} = 15\% \)

3. P(student has cat OR dog):

Total with at least one pet = 15 + 20 + 35 = 70

\( P(\text{cat or dog}) = \frac{70}{100} = 70\% \)

4. P(student has a cat GIVEN they have a dog): (Conditional probability)

\( P(\text{cat}|\text{dog}) = \frac{15}{50} = \frac{3}{10} = 30\% \)

5. Make Predictions Using Probability

Goal: Use probability to predict how many times an event will occur in a certain number of trials.

Formula:

\( \text{Expected number} = P(\text{event}) \times \text{Number of trials} \)

Steps:

Find the probability of the event
Multiply by the number of trials
Round to a whole number if necessary

Examples:

Example 1: If you roll a die 60 times, how many times would you expect to roll a 3?

\( P(3) = \frac{1}{6} \)

Expected: \( \frac{1}{6} \times 60 = 10 \) times

Example 2: A basketball player makes 70% of free throws. In 40 attempts, how many are expected to go in?

Expected: \( 0.70 \times 40 = 28 \) free throws

Example 3: In a bag with 5 red and 3 blue marbles, if you draw 200 times (with replacement), how many blue marbles would you expect?

\( P(\text{blue}) = \frac{3}{8} \)

Expected: \( \frac{3}{8} \times 200 = 75 \) blue marbles

6. Compound Events: Find the Number of Outcomes

Definition: A compound event is an event that involves two or more simple events.

Methods to Count Outcomes:

1. Tree Diagram: Branch out all possibilities visually

2. List/Table: Write out all possible outcomes

3. Fundamental Counting Principle: Multiply the number of outcomes for each event

Example:

Flip a coin and roll a die. How many possible outcomes?

Coin outcomes: 2 (H or T)

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

Total outcomes: 2 × 6 = 12

List: H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6

7. Compound Events: Find the Number of Sums

Common Problem: Rolling two dice and finding probabilities of different sums.

Two Dice Sum Table:

Die 1 ↓ / Die 2 →	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9

Key Facts:

Total outcomes: 6 × 6 = 36
Possible sums: 2 through 12
Most common sum: 7 (6 ways)
Least common sums: 2 and 12 (1 way each)

Example:

P(sum = 7) when rolling two dice:

Ways to get 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 ways

\( P(\text{sum}=7) = \frac{6}{36} = \frac{1}{6} \)

8. Independent and Dependent Events

Independent Events:

Definition: The outcome of one event does NOT affect the outcome of the other.

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

Example: Flipping a coin and rolling a die
Example: Drawing with replacement (put back and redraw)
Example: Weather on different days

Dependent Events:

Definition: The outcome of one event DOES affect the outcome of the other.

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

Example: Drawing without replacement (don't put back)
Example: Choosing 2 people from a group

How to Identify:

Ask yourself:	Answer
Does first event change the second?	Yes → Dependent
Is there replacement?	Yes → Independent; No → Dependent
Are events completely separate?	Yes → Independent