Chance Carnival Reference Sheet
Math Quest
Chance Carnival
The Ultimate Guide to Middle School Probability (Grades 6-8)
Name: ________________________
Date: _________ Period: ______
The Probability Scale (Certainty Meter)
All probability values range from 0 to 1
Impossible Value: 0 (0%) E.g., Rolling an 8 on a standard die.
Unlikely 1/4 | 0.25 | 25% Picking 1 red marble from 4 total.
Equally Likely 1/2 | 0.50 | 50% Flipping "Heads" on a fair coin.
Likely 3/4 | 0.75 | 75% Choosing a school day from a week.
Certain Value: 1 (100%) Rolling a number < 7 on a standard die.
The Probability Formula
Calculates the probability of a single simple event occurring.
Theoretical Formula
\[ P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Possible Outcomes}} \]
EXAMPLE: Roll a standard 6-sided die. What is the probability of rolling a 5 or 6?
Favorable: 2 ({5, 6}) Total: 6 ({1-6})
\( P(\text{5 or 6}) = \frac{2}{6} = \frac{1}{3} \approx 0.333 \approx 33.3\% \)
Key Terms to Master
Experiment An action, trial, or process with uncertain outcomes. (E.g., spinning a spinner).
Outcome A single possible result of an experiment. (E.g., landing on Blue).
Sample Space The set of ALL possible outcomes. (E.g., {Red, Yellow, Blue}).
Event A specific outcome or collection of outcomes. (E.g., spinning a primary color).
The Probability Showdown
Understanding the difference between expectation and reality.
CASE STUDY: THE MARBLE BAG
A closed bag contains 3 Red Marbles and 2 Blue Marbles (Total = 5).
Theoretical Probability "In Theory"
What should happen in a perfect mathematical world, calculated without running trials.
\( P(\text{Red}) = \frac{3 \text{ Red Marbles}}{5 \text{ Total Marbles}} = 0.60 = 60\% \)
Experimental Probability "In Practice"
What actually happens when you run trials, record outcomes, and analyze real data.
Suppose we draw with replacement 50 times:
Result: Drew Red 28 times and Blue 22 times.
\( \text{Exp. } P(\text{Red}) = \frac{28 \text{ Red Draws}}{50 \text{ Total Trials}} = 0.56 = 56\% \)
THE LAW OF LARGE NUMBERS: As you perform more trials, your Experimental Probability will get closer and closer to your Theoretical Probability!
Mapping Outcomes for Compound Events
Compound events combine two or more simple events. Here are two main ways to list their outcomes.
Double Challenge
You flip a coin (H/T) AND spin a spinner with 3 colors (Red, Blue, Green).
Counting Principle
Multiply options:
2 × 3 = 6 Outcomes
Method 1: Systematic Table
Coin
Spinner
Outcome
Heads (H)
Red (R)
H - R
Heads (H)
Blue (B)
H - B
Heads (H)
Green (G)
H - G
Tails (T)
Red (R)
T - R
Tails (T)
Blue (B)
T - B
Tails (T)
Green (G)
T - G
Method 2: Visual Tree Diagram
Heads (H)
─┬─
Red (R) ➔ H-R
Blue (B) ➔ H-B
Green (G) ➔ H-G
Tails (T)
─┬─
Red (R) ➔ T-R
Blue (B) ➔ T-B
Green (G) ➔ T-G
