How to Explain Probability to Eighth Graders

"What are the odds of that happening?"

We ask this question constantly—about games, weather, sports, and life decisions. Probability gives us a mathematical framework for understanding uncertainty and making informed choices in an unpredictable world.

Why Probability Matters

Probability is used in:

• Weather forecasting
• Medical testing and diagnosis
• Insurance and risk assessment
• Games and gambling
• Sports analytics
• Business decisions
• Scientific research

Probability Basics Review

What Is Probability?

Probability measures how likely an event is to occur, expressed as a number from 0 to 1.

0                 0.5                 1
|----------|--------|----------|---------|
Impossible   Unlikely   Equally    Likely    Certain
                       Likely

The Probability Formula

P(event) = Number of favorable outcomes
           ----------------------------
           Total number of possible outcomes

Example: Rolling a Die

P(rolling a 4) = 1/6

• Favorable outcomes: {4}
• Total outcomes: {1, 2, 3, 4, 5, 6}

Probability Notation

• P(A) = probability of event A
• P(not A) = P(A') = probability A doesn't happen
• P(A and B) = probability both A and B happen
• P(A or B) = probability A or B (or both) happen

Theoretical vs. Experimental Probability

Theoretical Probability

What SHOULD happen based on mathematical analysis.

P(heads) = 1/2 = 0.5 = 50%

(assuming a fair coin)

Experimental Probability

What ACTUALLY happens when you run trials.

Flip a coin 100 times, get 47 heads.

P(heads) = 47/100 = 0.47 = 47%

Law of Large Numbers

As the number of trials increases, experimental probability approaches theoretical probability.

10 flips:    60% heads (6/10)
100 flips:   52% heads (52/100)
1000 flips:  50.3% heads (503/1000)
10000 flips: 49.98% heads (4998/10000)

Compound Events

Independent Events

Events that don't affect each other.

Example: Rolling two dice

• What you roll on the first die doesn't affect the second

Dependent Events

Events where one outcome affects the next.

Example: Drawing cards without replacement

• After drawing an ace, there are fewer aces and fewer cards total

The Multiplication Rule (AND)

For Independent Events

P(A and B) = P(A) × P(B)

Example: Rolling two 6s in a row

P(6 on first) × P(6 on second)
= (1/6) × (1/6)
= 1/36

For Dependent Events

P(A and B) = P(A) × P(B|A)

Where P(B|A) = probability of B given that A happened.

Example: Drawing two aces from a deck (without replacement)

P(1st ace) = 4/52
P(2nd ace | 1st ace) = 3/51 (one ace gone, one card gone)

P(both aces) = (4/52) × (3/51) = 12/2652 = 1/221

The Addition Rule (OR)

Mutually Exclusive Events

Events that CAN'T happen at the same time.

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

Example: Rolling a 5 or 6 on one die

P(5 or 6) = P(5) + P(6) = 1/6 + 1/6 = 2/6 = 1/3

Non-Mutually Exclusive Events

Events that CAN happen together.

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

Example: Drawing a heart or a king

P(heart) = 13/52
P(king) = 4/52
P(heart AND king) = 1/52 (king of hearts)

P(heart or king) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13

Why Subtract?

We subtract P(A and B) to avoid counting the overlap twice!

Hearts:  ♥A ♥2 ♥3 ♥4 ♥5 ♥6 ♥7 ♥8 ♥9 ♥10 ♥J ♥Q ♥K
Kings:   ♠K ♥K ♦K ♣K

♥K is in BOTH lists, so we subtract it once.

Tree Diagrams

Visualizing Compound Events

Tree diagrams show all possible outcomes and their probabilities.

Example: Flipping a coin twice

                Start
               /     \
            H(1/2)   T(1/2)
            /  \      /  \
         H     T    H     T
        1/4   1/4  1/4   1/4

Outcomes: HH, HT, TH, TT (each 1/4)

Example: Drawing Marbles

Bag has 3 red, 2 blue. Draw 2 without replacement.

                  Start
               /        \
           R(3/5)      B(2/5)
           /    \       /    \
       R(2/4)  B(2/4) R(3/4) B(1/4)
         |       |      |      |
       RR      RB     BR     BB
       6/20   6/20   6/20   2/20

P(both red) = 3/5 × 2/4 = 6/20 = 3/10
P(both blue) = 2/5 × 1/4 = 2/20 = 1/10
P(one of each) = 6/20 + 6/20 = 12/20 = 3/5

Sample Spaces and Counting

Listing Outcomes

For small experiments, list all outcomes systematically.

Example: Rolling two dice

     1   2   3   4   5   6
  +------------------------
1 | 1,1 1,2 1,3 1,4 1,5 1,6
2 | 2,1 2,2 2,3 2,4 2,5 2,6
3 | 3,1 3,2 3,3 3,4 3,5 3,6
4 | 4,1 4,2 4,3 4,4 4,5 4,6
5 | 5,1 5,2 5,3 5,4 5,5 5,6
6 | 6,1 6,2 6,3 6,4 6,5 6,6

Total outcomes: 36

Counting Principle

If event A can happen m ways and event B can happen n ways, then A and B together can happen m × n ways.

Example: Outfits from 4 shirts and 3 pants

4 × 3 = 12 possible outfits

Simulations

What Is a Simulation?

Using random number generators or physical tools to model probability experiments.

Why Simulate?

• Real experiments might be impractical (too expensive, too slow)
• Theoretical probability might be too complex to calculate
• To verify theoretical predictions

Simulation Tools

• Coins, dice, spinners
• Random number generators (calculators, computers)
• Random number tables

Example Simulation

Question: In a family with 3 children, what's the probability of having exactly 2 girls?

Simulation design:

• Let 0 = boy, 1 = girl
• Generate 3 random digits (0 or 1)
• Count successes (exactly two 1s)

Trial results (20 trials):

011, 101, 110, 000, 111, 010, 100, 011, 110, 001
100, 111, 010, 101, 110, 011, 000, 110, 101, 011

Exactly 2 girls: 12 times
Experimental P = 12/20 = 0.60 = 60%

Theoretical P = 3/8 = 0.375 = 37.5%

(More trials would bring experimental closer to theoretical)

Probability Distributions

What They Show

A probability distribution lists all possible outcomes with their probabilities.

Example: Sum of two dice

Sum	2	3	4	5	6	7	8	9	10	11	12
P	1/36	2/36	3/36	4/36	5/36	6/36	5/36	4/36	3/36	2/36	1/36

Properties

• All probabilities are between 0 and 1
• All probabilities sum to 1

Visualizing with a Histogram

P(sum)
  |
1/6+       _____
  |       |     |
5/36+    _|     |_
  |     |         |
4/36+  _|         |_
  |   |             |
3/36+ _|             |_
  |  |                 |
2/36+_|                 |_
  | |                     |
1/36|_                     _|
  +--2--3--4--5--6--7--8--9--10--11--12
                         Sum

Expected Value

What Is It?

The expected value is the average outcome over many trials.

Formula

E(X) = Σ [outcome × probability]

Example: Dice Roll

E(X) = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6)
E(X) = (1+2+3+4+5+6)/6
E(X) = 21/6 = 3.5

On average, you expect to roll 3.5 (even though you can never actually roll 3.5!)

Game Fairness

A game is fair if the expected value is 0 (neither player has an advantage).

Example: Coin flip game

• Win $1 on heads
• Lose $1 on tails

E(X) = 1(0.5) + (-1)(0.5) = 0

Fair game!

Hands-On Activities

Coin Flip Experiment

• Flip a coin 50 times
• Record heads/tails
• Calculate experimental probability
• Compare to theoretical (0.5)
• Combine class data—what happens?

Dice Sum Investigation

• Roll two dice 36 times
• Record each sum
• Create a frequency table
• Compare to theoretical distribution
• Which sum appeared most?

Design a Fair Game

Create a game using dice, coins, or spinners:

• Calculate expected value for each player
• Adjust rules until the game is fair
• Test with actual play

Monte Hall Simulation

The famous "three doors" problem:

• 3 doors: 1 prize, 2 goats
• Pick a door
• Host opens a goat door
• Should you switch?

Simulate to find the surprising answer!

Real-World Probability Hunt

Find probability statements in the news:

• Weather forecasts
• Sports predictions
• Medical statistics

What do the numbers mean?

Common Mistakes and How to Fix Them

Mistake 1: Multiplying When You Should Add

Wrong: P(5 or 6) = (1/6) × (1/6) = 1/36

Fix: "Or" means add for mutually exclusive events.
P(5 or 6) = 1/6 + 1/6 = 2/6 = 1/3

Mistake 2: Forgetting Probability Changes (Dependent Events)

Wrong: P(two aces) = (4/52) × (4/52)

Fix: After drawing first ace, only 3 aces and 51 cards remain.
P(two aces) = (4/52) × (3/51) = 1/221

Mistake 3: The Gambler's Fallacy

Wrong: "I've flipped 5 heads in a row, so tails is 'due'"

Fix: Coins have no memory! Each flip is independent. P(tails) is still 1/2.

Mistake 4: Confusing "At Least" with "Exactly"

Wrong: P(at least one head in 2 flips) = 1/4

Fix: "At least one" includes HH, HT, TH = 3 outcomes.
P(at least one head) = 3/4

Or use complement: P(at least one H) = 1 - P(no H) = 1 - 1/4 = 3/4

Mistake 5: Double Counting in "Or" Problems

Fix: For non-mutually exclusive events, subtract the overlap!
P(A or B) = P(A) + P(B) - P(A and B)

Practice Ideas for Home

Probability Games

Play games and calculate odds:

• Board games with dice
• Card games
• Probability-based video games

Sports Probability

Analyze your favorite sport:

• What's a player's free throw percentage?
• What's the probability of a hit?
• Does the home team really have an advantage?

Weather Tracking

Track weather forecasts:

• When they say "70% chance of rain," how often does it rain?
• Keep a log and calculate accuracy

Design Experiments

Create and run probability experiments:

• Make a weighted spinner
• Calculate theoretical vs experimental probability
• How many trials until they're close?

Connecting to Future Concepts

Binomial Probability

What's the probability of exactly k successes in n trials?

P(X = k) = C(n,k) × p^k × (1-p)^(n-k)

Normal Distribution

The bell curve describes many real-world phenomena.

Conditional Probability and Bayes' Theorem

More sophisticated analysis of dependent events.

Statistics and Inference

Using probability to make conclusions from sample data.

Actuarial Science

Insurance companies use probability to set rates and assess risk.

The Bottom Line

Probability quantifies uncertainty, allowing us to make better decisions in an unpredictable world.

Key concepts for eighth graders:

• Theoretical vs. experimental probability—what should happen vs. what does happen
• Compound events—using AND (multiply) and OR (add)
• Independent vs. dependent events—does one outcome affect the next?
• Tree diagrams and counting—visualizing all possibilities
• Simulations—modeling complex situations

When students understand probability, they can:

• Evaluate claims and risks critically
• Make informed decisions under uncertainty
• See through misleading statistics
• Appreciate the mathematics behind games and gambling

Probability thinking is essential for navigating a world full of uncertainty—from weather forecasts to medical decisions to everyday choices.