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Interactive Probability & Law of Large Numbers Activity for Grades 7-10
In this interactive lesson, you'll explore how probability works in the real world through hands-on simulations. Learn why scientists, game designers, and statisticians run many trials to get accurate results!
Real-World Context: Gaming, Science
Math Concepts: Probability, Law of Large Numbers
Have you ever wondered why scientists repeat their experiments many times? Or why game designers test their games over and over? The answer lies in probability and something called the Law of Large NumbersA principle that states that as a process is repeated more times, the actual outcomes will converge on the expected theoretical probability..
Think About: If you flip a fair coin 10 times, would you always get exactly 5 heads and 5 tails? Why or why not?
Probability: The likelihood that a specific outcome will occur. It ranges from 0 (impossible) to 1 (certain).
Law of Large Numbers: As you increase the number of trials (like coin flips or dice rolls), the actual results will get closer and closer to the expected probability.
The chart above shows what typically happens when you flip a fair coin many times. Notice how with just a few flips, the percentage of heads can vary widely. But as you do more and more flips, the percentage gets closer to the expected 50%.
Why this matters: In the real world, understanding probability and the Law of Large Numbers helps us:
Let's start with a simple experiment: flipping a fair coin. Theoretically, a fair coin has a 50% chance of landing on heads and a 50% chance of landing on tails.
Total flips: 0
Heads: 0 (0%)
Tails: 0 (0%)
Questions to Consider:
With just a few flips, the results can vary widely from the expected 50% probability. This is normal! Random events don't always follow the expected pattern in small samples.
But as you increase the number of flips, something interesting happens: the actual percentage gets closer and closer to the theoretical 50%. This is the Law of Large Numbers in action.
Now let's try something a bit more complex: rolling a six-sided die. Each face has a 1/6 (about 16.7%) probability of showing up on any single roll.
Total rolls: 0
| Face | Count | Percentage | Expected |
|---|---|---|---|
| 1 | 0 | 0% | 16.7% |
| 2 | 0 | 0% | 16.7% |
| 3 | 0 | 0% | 16.7% |
| 4 | 0 | 0% | 16.7% |
| 5 | 0 | 0% | 16.7% |
| 6 | 0 | 0% | 16.7% |
Questions to Consider:
With a die, you have six possible outcomes instead of just two, so it can take even more rolls to see the expected distribution. This is why games that use dice often require multiple rolls or multiple dice - to smooth out the randomness.
This is also why scientists don't rely on just a few trials for their experiments. They need many data points to be confident that what they're observing isn't just due to random chance.
Now let's try a weighted probability experiment. This spinner has four sections, but they're not equal in size. The larger sections have a higher probability of being selected.
Total spins: 0
| Section | Size | Expected % | Count | Actual % |
|---|---|---|---|---|
| Blue (Large) | 50% | 50% | 0 | 0% |
| Light Blue | 25% | 25% | 0 | 0% |
| Dark Blue | 12.5% | 12.5% | 0 | 0% |
| White | 12.5% | 12.5% | 0 | 0% |
Questions to Consider:
This experiment shows something important: outcomes with smaller probabilities often need more trials to reach their expected percentages. This is why rare events or small effects in science require larger sample sizes to detect accurately.
Think about testing a new medicine that helps 5% of patients. If you only test it on 10 people, you might not see any effect at all! But if you test it on 1,000 people, you're likely to see close to 50 people helped, which gives you much more confidence in the results.
Now it's your turn to design an experiment! Your goal is to find out how many trials are needed to get reliable results for different probability scenarios.
Choose a probability experiment and decide how many trials to run. Then predict what you think will happen!
Make your prediction:
How close to the theoretical probability do you think your results will be?
Your experiment results will appear here after you run the experiment.
Different scenarios require different numbers of trials to get reliable results:
| Scenario | Approximate Trials Needed for Reliable Results |
|---|---|
| 50/50 probability (coin flip) | 100+ trials to get within 5% of expected |
| 1 in 6 probability (dice roll) | 300+ trials to get within 5% of expected |
| Rare events (5% probability) | 500+ trials to detect reliably |
| Very rare events (1% probability) | 1000+ trials to detect reliably |
Think About: Why do rarer events require more trials? How might this apply to situations like testing a rare side effect of a medication?
Now that you've experimented with probability and the Law of Large Numbers, let's think about how these concepts apply in the real world.
Discussion Prompts:
Game designers need to understand probability to:
For example, if a game has a 1% chance to drop a rare item, players might need to defeat hundreds of enemies before finding one. Understanding this helps designers determine if that rate feels fair and fun.
Scientists rely on probability and large samples to:
Clinical trials often need thousands of participants to reliably detect both beneficial effects and rare side effects of new treatments.
Write a short reflection on what you've learned about probability and the Law of Large Numbers. Consider these questions:
Want to learn more? Try these activities:
Understanding probability and the Law of Large Numbers is essential for careers in:
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