Introduces probability as a measure from 0 to 1 representing likelihood. Students list sample spaces for simple and compound events, calculate theoretical probability using equally likely outcomes, conduct experiments to find experimental (relative frequency) probability, and compare the two as trial counts increase (law of large numbers intuition).
Middle School Geometry & Data • 6-8
Probability measures how likely an event is to happen. It is a number between 0 (impossible) and 1 (certain). Understanding probability helps you make predictions, evaluate risk, and think critically about chance in games, science, and everyday decisions.
Probability can be written as a fraction, decimal, or percent. For example, 1/4 = 0.25 = 25%.
A sample space is the set of all possible outcomes of an experiment.
Simple Events:
Compound Events:
For compound events, multiply the number of outcomes in each individual experiment (this is the counting principle).
This formula works when all outcomes are equally likely. You calculate it by reasoning, without running an experiment.
This is based on actual data from performing the experiment. It may differ from the theoretical probability, especially with few trials.
As you perform more and more trials, the experimental probability gets closer and closer to the theoretical probability. Flip a coin 10 times and you might get 70% heads. Flip it 10,000 times and you will get very close to 50%. This pattern is called the Law of Large Numbers.
What is the probability of rolling a number greater than 4 on a standard die?
You flip a coin and roll a die. What is the probability of getting Heads and a 3?
A spinner has 4 equal sections: red, blue, green, yellow. You spin 50 times and get red 16 times. Compare experimental and theoretical probability of red.
For compound events, use an organized list, a tree diagram, or a table to avoid missing outcomes. For two coins: list HH, HT, TH, TT. Note that HT and TH are different outcomes (first coin vs. second coin), so do not combine them.
Students sometimes think that after flipping 5 heads in a row, tails is "due." Each flip is independent—past results do not change future probabilities. The coin has no memory. P(Tails) is always 1/2 on any single flip.
1. A bag holds 3 red, 5 blue, and 2 green marbles. What is the probability of drawing a blue marble?
Total = 3 + 5 + 2 = 10. P(blue) = 5/10 = 1/2 = 50%.
2. List the sample space for flipping three coins. How many outcomes are there?
{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} — 8 outcomes (2 × 2 × 2 = 8).
3. Using the sample space from Problem 2, what is P(exactly 2 heads)?
Favorable: {HHT, HTH, THH} = 3 outcomes. P = 3/8 = 0.375 = 37.5%.
4. A student rolls a die 60 times and gets a 6 exactly 8 times. (a) What is the experimental probability? (b) What is the theoretical probability? (c) Are they close?
(a) Experimental: 8/60 = 2/15 ≈ 13.3%. (b) Theoretical: 1/6 ≈ 16.7%. (c) Reasonably close; with more trials the experimental probability would likely approach 16.7%.
5. An event has probability 0. What does this mean? What about probability 1?
Probability 0 means the event is impossible—it cannot happen. Probability 1 means the event is certain—it must happen. All probabilities fall between these extremes.