Learning from experiments
Experimental probability comes from actually doing something many times and recording what happens.
P(event) = number of times the event happens ÷ total number of trials
Each repetition is a trial. The count of how often an outcome happens is its frequency, so experimental probability is also called relative frequency.
Example: flipping a coin
You flip a coin 50 times and get 28 heads.
P(heads) = 28/50 = 0.56 (56%)
Example: rolling a die
| Outcome | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Frequency | 9 | 11 | 8 | 12 | 10 | 10 |
Total trials: 60.
P(rolling 4) = 12/60 = 0.2
P(even number) = (11 + 12 + 10)/60 = 33/60 = 0.55
Experimental vs theoretical
| Experimental | Theoretical | |
|---|---|---|
| Based on | Actual results | Reasoning about equally likely outcomes |
| Coin: P(heads) | e.g. 0.56 after 50 flips | exactly 0.5 |
| Changes? | Yes, with each experiment | No |
The law of large numbers
Short experiments can give surprising results. Flip a coin 10 times and you might get 7 heads (70%). But as the number of trials grows, the experimental probability gets closer to the theoretical value. This is the law of large numbers.
| Flips | Heads | P(heads) |
|---|---|---|
| 10 | 7 | 0.70 |
| 100 | 54 | 0.54 |
| 1,000 | 507 | 0.507 |
| 10,000 | 4,988 | 0.4988 |
When experiments are the only way
Some probabilities can’t be worked out by reasoning:
- The chance a thumbtack lands point up
- The chance a basketball player makes a free throw
- The chance a light bulb lasts over 1,000 hours
For these, collect data. A player who made 78 of 100 free throws has an experimental probability of 0.78.
Making predictions
Use experimental probability to predict.
Example: A factory finds 3 faulty items in a sample of 200. How many faulty items might be in 5,000?
3/200 × 5,000 = 75
Worked examples
Example 1: A spinner lands on blue 18 times in 60 spins. P(blue)?
18/60 = 0.3.
Example 2: It rained on 9 of the last 30 days. Experimental probability of rain?
9/30 = 0.3 (30%).
Example 3: Is 0.56 for heads unusual after 50 flips?
Not really; small experiments vary.
Common mistakes
- Expecting experimental and theoretical results to match exactly.
- Drawing big conclusions from very few trials.
- Dividing by the wrong total.
Try it yourself
- 40 flips, 22 tails. P(tails)?
- A die is rolled 120 times; 6 comes up 25 times. Experimental P(6)? Theoretical P(6)?
- A goalie saves 45 of 60 shots. Predict saves out of 100 shots.
- Why do more trials give better estimates?
Answers: 1) 0.55 2) 25/120 ≈ 0.21; 1/6 ≈ 0.17 3) 75 4) Random variation evens out (law of large numbers).
Quick recap
- Experimental probability = event count ÷ trials.
- It’s also called relative frequency.
- More trials bring it closer to the theoretical probability.