Probability: What Should Happen and What Actually Did
A 45-minute math lesson on probability — writing likelihood as a number from 0 to 1, separating what theory predicts from what an experiment actually produced, and why a coin is never 'due' for the other side.
Math
Subject
Grades 6-8
Grade Level
45 minutes
Duration
Probability
Topic
On this page: Materials · Warm-Up · Direct Instruction · Guided Practice · Independent Practice · Assessment · Closure · Related Resources
Learning Objectives
By the end of this lesson, students will be able to:
- Write the probability of a simple event as a fraction, decimal, or percent between 0 and 1
- Calculate theoretical probability from the structure of a situation
- Calculate experimental probability from the results of trials that were actually run
- Explain why theoretical and experimental probability get closer as the number of trials grows
- Explain why an independent event is not affected by the results before it
Materials
- Theoretical and Experimental Probability Worksheet — one copy per student
- Probability of Compound Events Worksheet and Simple Probability Worksheet, for extension
- One coin per pair, plus a number cube per pair
- A board or chart for pooling the whole class's trial results
Vocabulary
- Probability — a number from 0 to 1 describing how likely an outcome is
- Outcome — one possible result of a trial, such as landing on heads
- Theoretical probability — what the structure of a situation predicts: favorable outcomes divided by total possible outcomes
- Experimental probability — what actually happened: times the outcome occurred divided by trials run
- Independent events — events where the result of one does not change the probability of the next
Preparation
Have one coin and one number cube per pair, and draw a pooling table on the board with columns for each pair’s heads count and total flips — the whole-class total is what makes the convergence visible, and it needs somewhere to live.
Decide in advance how many flips each pair will run (ten is enough to produce a messy result, which is the point) and work out roughly what the class total will be, so you are not doing that arithmetic in front of them.
Be ready for the class pooled result to land near half but not exactly on it. That is the honest outcome and the lesson is built on it — do not massage the numbers toward 50 percent.
Warm-Up
4 minutes. Tell the class a coin has just landed heads five times in a row, and ask for a show of hands: is the next flip more likely to be tails, more likely to be heads, or equally likely?
Most classes will vote heavily for tails. Do not correct it yet. Ask the students who voted tails to say WHY in their own words — the word "due" will usually come up, and it is worth writing on the board so the class can test it later.
Say plainly where the lesson is going: today the class will run its own trials and then use them to work out what "due" would have to mean. Do not promise that the tally by itself will answer it — no run of flips can prove a coin has no memory, and the class will need the structure of the coin as well as its own numbers to get there.
Direct Instruction
13 minutes. Define probability as a number from 0 to 1: 0 means it cannot happen, 1 means it must happen, and everything real sits between. Point out that a probability can be written as a fraction, a decimal, or a percent, and that all three say the same thing.
Define theoretical probability: favorable outcomes divided by total possible outcomes, worked out from the STRUCTURE of the situation without touching a coin. A number cube has six equally likely faces, so rolling a 4 is 1/6. Stress the phrase "equally likely" — the formula only works when every outcome has the same chance.
Define experimental probability: the number of times the outcome actually happened divided by the number of trials actually run. Make the contrast explicit — theoretical is a prediction from structure, experimental is a report from data, and they are answers to two different questions.
Now run the experiment. Each pair flips ten times and records heads. Collect the pair totals on the board, then compute the experimental probability twice: once for a single pair, and once for the pooled class total. The single pair will usually be well off 0.5; the pooled total will usually be close to it.
Read the two figures the class actually produced rather than the ones you expected. In most classes the pooled total sits closer to 0.5 than the single pair did, and when it does, name it: more trials pulled the experimental result toward the theoretical one. When it does not — and sometimes it will not — that is the better lesson and worth taking. Convergence is a long-run tendency, not a promise that each new batch of flips lands closer than the last one did. Either way, then define independent events and use it to settle the warm-up: the coin has no memory, so each flip stays at 1/2 regardless of what came before. The long-run pattern is not restored by the coin correcting itself — it is restored by later results swamping the early ones. Cross "due" off the board.
Guided Practice
13 minutes. In pairs, students compute the theoretical probability of several number-cube events before rolling anything: rolling a 5, rolling an even number, rolling a number greater than 4.
Pairs then roll twenty times, record results, and compute the experimental probability for the same three events. They write one sentence comparing each pair of numbers.
Circulate with one question: which of these two numbers would change if you rolled twenty more times, and which one could not change at all? The theoretical value is fixed by the cube; only the experimental value moves.
Pairs then start the Theoretical and Experimental Probability worksheet together.
Independent Practice
10 minutes. Students complete the Theoretical and Experimental Probability worksheet.
Add one written item: a spinner has landed on red four times in a row. A classmate says red is "on a streak" and another says red is "used up." Explain in two or three sentences why both classmates are making the same mistake.
Timing note. The worksheet will often fill the whole ten minutes on its own. When it does, run the added item as homework or as the start of the next session rather than compressing both — the exit ticket needs its two minutes, and a rushed version of this task is worth less than no version of it.
Assessment
2 minutes. Exit ticket, two items. A bag holds 3 red and 5 blue marbles — what is the theoretical probability of drawing red? Then: a class flipped a coin 20 times and got 13 heads. Is the coin unfair? What would you do before deciding?
The second item is the one that discriminates. The first is a formula check; the second requires knowing that a short run of trials is expected to wander from the theoretical value, so the honest answer is to run many more flips before concluding anything.
Closure
3 minutes. Put the single-pair result and the pooled class result back on the board side by side and have the class say what the difference between them shows. Close on the rule: theoretical probability tells you what to expect in the long run, experimental probability tells you what happened this time, and no individual trial is ever owed a particular result.
Differentiation and Accommodations
- Extra support: stay with theoretical probability on a single number cube for today, writing each answer as a fraction only, and save the experimental half for a second lesson. Reliably counting favorable outcomes over total outcomes is a real result on its own.
- Extension: the probability of compound events worksheet (grades 7-8) moves to two events at once, which is the natural next step, and the simple probability worksheet (grades 5-7) gives lighter practice on single events.
- Common difficulty: a student who writes a probability greater than 1, which almost always means they divided total by favorable instead of the other way round. Ask what a probability of 3 would even mean — the impossible answer is the fastest route back to the right order.
- Watch for: a student who concludes from the pooled result that the coin "proved" probability is exactly 1/2. It did not, and saying so matters: more trials make the experimental value close in on the theoretical one without ever being guaranteed to land on it.
Extension Activities
Have students run 100 trials at home with a coin and bring the tally in, then pool the whole class again — a few thousand flips gets strikingly close to half, and seeing that scale makes the long-run argument concrete in a way ten flips cannot.
Connect to Ratios and Proportional Relationships: a probability is a ratio of favorable outcomes to total outcomes, so students who can already reduce and compare ratios are doing the same arithmetic in a new setting.
For a class that wants to test "due" directly rather than settle it by argument, have students go back through their own recorded flips and look only at the flips that came immediately AFTER a heads. With enough of them the share of heads in that subset tends toward half too, which is the closest a tally can come to showing the coin is not keeping score — though a class-sized subset is small, so expect it to be noisy and do not present whatever number appears as the answer. Say plainly what the exercise can and cannot do: the class looked for the effect and did not find it, which is weaker and more honest than proof.
Pose a research question: find a real weather forecast that gives a percent chance of rain, and work out what a forecaster actually means by "a 30 percent chance" — it is a statement about many similar days, not about tomorrow alone.
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