🔄 Grade 7 · Lesson 11 of 14
Experimental vs. Theoretical Probability
Calculate what should happen, then flip a coin and see what actually does — and watch the gap between them shrink the more you flip.
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The big idea: Theoretical probability is calculated ahead of time by reasoning about equally likely outcomes, like knowing a fair coin lands heads with probability 1/2. Experimental probability, by contrast, comes from actually running trials and counting results. With only a few trials, experimental results can differ noticeably from the theoretical prediction — but the more trials you run, the closer the experimental probability tends to settle toward the theoretical one.
Two ways to think about chance
Flip a coin, roll a die, spin a spinner — there are two different ways to talk about how likely an outcome is. One is to reason it out in advance, using logic about equally likely outcomes. The other is to actually try it a bunch of times and see what happens. They usually agree — but not always exactly, and understanding why is the point of this lesson.
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Theoretical probability: reasoned in advance
Theoretical probability is calculated by reasoning about equally likely outcomes, before any trial happens: divide the number of favorable outcomes by the total number of equally likely outcomes.
\[ P(\text{event}) = \dfrac{\text{number of favorable outcomes}}{\text{number of equally likely outcomes}} \]
Theoretical probability, based purely on reasoning about the possibilities.
Experimental probability: measured from data
Experimental probability instead comes from actually running trials — flipping the coin, rolling the die, spinning the spinner — and counting how often the event actually happened out of the total number of trials.
\[ P_{\text{experimental}}(\text{event}) = \dfrac{\text{number of times the event occurred}}{\text{total number of trials}} \]
Experimental probability, based on real, observed results.
🎮 Million-Coin Flipper LIVE
Flip thousands of coins and watch the experimental probability settle toward the theoretical 1/2.
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More trials, closer match
With only a handful of trials, experimental probability can swing noticeably away from the theoretical value — getting 4 heads out of 5 flips does not mean the coin is unfair. But as the number of trials grows into the hundreds or thousands, the experimental probability reliably settles closer and closer to the theoretical value.
📝 Worked example: A fair coin is flipped once. What is the theoretical probability of getting heads?
- There are 2 equally likely outcomes: heads and tails.
- Exactly 1 of those outcomes (heads) is favorable.
- P(heads) = 1 ÷ 2.
✓ The theoretical probability of heads is <strong>1/2</strong> (50%).
📝 Worked example: You flip a coin 20 times and get 12 heads. What is the experimental probability of heads, and how does it compare to the theoretical probability?
- Experimental probability = number of heads ÷ total flips = 12 ÷ 20.
- Simplify: 12/20 = 3/5 = 0.6.
- Compare to the theoretical probability of 1/2 = 0.5.
✓ The experimental probability is <strong>3/5, or 60%</strong> — somewhat higher than the theoretical 50%, a normal result of chance over a fairly small number of trials.
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A short streak doesn't mean the coin is unfair
Getting an unusual result over a small number of trials — like 4 heads in 5 flips — does not mean the coin is biased. Small samples naturally vary. Only a large number of trials that consistently stays far from 1/2 would suggest something is actually wrong with the coin.
Check your understanding
1. A fair six-sided die is rolled once. What is the theoretical probability of rolling a 4?
There are 6 equally likely outcomes, and only one of them is a 4, so P(4) = 1/6.
2. You roll a die 60 times and get a '4' exactly 8 times. What is the experimental probability of rolling a 4?
Experimental probability = 8 ÷ 60 = 2/15 ≈ 0.133, close to but not exactly the theoretical 1/6 ≈ 0.167.
3. According to the law of large numbers, as the number of coin flips increases, the experimental probability of heads tends to…
With more trials, experimental probability reliably settles closer to the theoretical probability.
4. A bag has 3 red marbles and 7 blue marbles (10 total). What is the theoretical probability of picking a red marble at random?
There are 10 equally likely marbles, 3 of which are red, so P(red) = 3/10.
5. You flip a coin 5 times and get 5 heads in a row. What does this most likely indicate?
With only 5 trials, unusual streaks happen naturally and don't indicate an unfair coin.
✅ Key takeaways
- Theoretical probability is reasoned out in advance: favorable outcomes over total equally likely outcomes.
- Experimental probability is measured from actual trials: how often an event happened over the total number of trials.
- With few trials, experimental probability can differ noticeably from theoretical probability, just by chance.
- As the number of trials grows, experimental probability tends to settle closer to the theoretical value.
- A short unusual streak, like several heads in a row, does not by itself mean a coin or die is unfair.
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