Experimental Probability and Long-Run Frequency

Lesson 1 of 1

In this lesson:

• Compute probability from real data
• See what happens as trials increase
• Predict and compare experimental results

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. Define experimental probability as relative frequency of observed trials
2. Compute from data
3. Explain why more trials produce more reliable probability estimates
4. Predict approximate frequency given a theoretical probability
5. Compare experimental results to theoretical probabilities
6. Understand that experimental probability approaches theoretical in the long run

Is Seven Heads in Ten Flips Unusual?

You already know probability lives between 0 and 1.

• P near 0 → very unlikely
• P near 1 → very likely
• A fair coin has P(heads) = 1/2

But what if you actually flip and get 7 heads in 10 tries?

Experimental Probability: Measuring from Real Data

Theoretical probability → comes from analysis (equal outcomes, symmetry)

Experimental probability → comes from actually running the experiment

It is always a fraction or decimal — never just a count.

Spinner Results: Computing Experimental Probability

• Spun 50 times; Blue appeared 18 times
• Theoretical P(Blue) if 4 equal sections = 0.25 — different, but not alarming

Worked Example: Coin Flip Results

Setup: A student flips a coin 40 times. Heads appears 23 times.

Theoretical:

Is the coin unfair? With only 40 flips, this difference is normal variation.

Example: Spinner with Three Colors

Setup: A spinner is spun 120 times. Results:

Sum check: 42 + 38 + 40 = 120 → all probabilities sum to 1 ✓

Check-In: Compute the Experimental Probability

A die is rolled 30 times. The outcome "4" appears 8 times.

Find the experimental probability of rolling a 4.

Work it out — then advance for the answer.

Check-In Answer: Rolling a 4 in 30 Trials

Theoretical:

Difference of about 0.10 — with only 30 trials, this is normal variation.

No reason to suspect the die is biased — the sample is too small to conclude.

What Happens When We Run More Trials?

We know experimental probability fluctuates with small samples.

The question: Does it stabilize as we run more trials?

Spoiler: Yes — and the pattern is one of the most important ideas in probability.

Long-Run Relative Frequency: Convergence Toward ½

The proportion bounces early — then settles near 0.5 as flips grow.

Why the Proportion Stabilizes Over Time

Early flips: Extra heads push proportion far from 0.5.

Later flips: Same extra heads are a tiny fraction of many flips.

• 5 extra heads / 10 flips → proportion shifts by 0.50
• 5 extra heads / 500 flips → barely noticeable

Randomness gets diluted — proportion stabilizes.

Check-In: Does the Coin "Remember"?

After 10 flips, you have 7 heads (70% heads so far).

Question: Are the next 10 flips more likely to be tails, to "balance out"?

Think before the next slide…

The Gambler's Fallacy: Coins Have No Memory

The Gambler's Fallacy: Believing a random process "corrects itself" after unusual results.

• Each flip is independent — no memory of past flips
• After 10 tails, P(tails) next flip is still 1/2
• "Balancing out" happens through dilution, not correction

Future flips never compensate for past results.

Two Running Proportion Graphs Compared

• 20 flips: large swings, unreliable estimate
• 200 flips: stable, reliable estimate of theoretical P

Check-In: Which Graph Is More Reliable?

Two students graph running proportions:

• Student A: 20 flips, final proportion = 0.60
• Student B: 200 flips, final proportion = 0.55

Which result is a more reliable estimate of P(heads)?

From Proportion to Prediction: Two Directions

Two directions with probability and data:

Formula: Expected frequency

Predicting Frequency from Theoretical Probability

Formula: Expected frequency

This is a prediction, not a guarantee.

• Actual count will be close — probably not exactly equal
• Natural variation means results scatter around the prediction

Standard example: "predict roughly 200 threes in 600 rolls — probably not exactly 200"

Worked Example: Marble Bag Frequency Prediction

Setup: Bag has 3 red, 7 blue (10 total). Draw with replacement 200 times.

Actual result will be close to 60 — probably between 45 and 75.

Example: Die Roll and Thumbtack Prediction

Die: ; roll 180 times

Thumbtack: Estimated ; flip 250 times

Actual result: 115 point-ups → new estimate:

Check-In: Predict Then Update from Data

Setup: ; draw from a bag 160 times.

Part A: How many red draws should you expect?

Part B: You actually draw 55 reds. What is your new experimental estimate of P(red)?

Work both parts before the next slide.

Check-In Answers: Prediction and Updating

Part A: Expected reds

Part B: Experimental

The actual result (55) is noticeably above the predicted (40).

This suggests the true P(red) may be higher than 0.25 — or it's natural variation.

Comparing Experimental and Theoretical Probability

When experimental P differs from theoretical P, what does it mean?

It depends on the number of trials:

• Small sample: Large deviations are expected — be cautious
• Large sample: Large deviations are unusual — worth investigating

Three Scenarios: Is the Die Biased?

• 30 rolls, P(6)≈0.267 vs 0.167 → Caution — small sample, normal variation
• 600 rolls, P(6)=0.250 vs 0.167 → Investigate — large deviation with large n
• 100 flips, P(heads)=0.500 vs 0.500 → Consistent — matches theoretical exactly

Guided Practice: Evaluating a Probability Claim

Scenario: A student draws 8 cards from a standard deck and gets 6 red.

She concludes: "The probability of drawing red must be about 3/4."

Theoretical P(red) = 1/2.

Evaluate her claim. Is the evidence sufficient?

Consider: How many trials? What deviation is normal with 8 draws?

Guided Practice Answer: Insufficient Evidence

With only 8 trials: getting 6 reds happens fairly often just by chance.

Conclusion: Evidence is insufficient to reject the theoretical P = 0.5.

More trials needed — perhaps 80 or 800 — before the data is meaningful.

Key Takeaways from Today's Lesson

✓ — a proportion, never just a count

✓ Experimental P approaches theoretical P as trials increase

✓ Expected frequency — a prediction, not a guarantee

✓ Large deviations only matter with large sample sizes

Five Common Probability Misconceptions to Avoid

1. "Expected means exact" — P × n is a prediction, actual results will vary
2. "The coin will balance out" — each flip is independent; no memory
3. "6 rolls with no sixes proves the die is biased" — need many more trials
4. "Experimental = theoretical always" — they agree long-run, not in every experiment
5. "30 heads means P(heads) = 30" — probability is a proportion, 30/60 = 0.5

Preview: Developing Formal Probability Models

Coming up in 7.SP.C.7:

• Build formal probability models from theoretical analysis
• Compare model predictions to experimental results systematically
• Decide when a model is a good fit for real data

Today's experimental probability is the empirical foundation for those models.