Probability Experiments
Explore the difference between what we expect (theoretical probability) and what actually happens (experimental probability), and understand how sample space helps us list all possible outcomes.
Probability Basics
Probability measures how likely an event is to occur. It is expressed as a number from 0 (impossible) to 1 (certain). We can also write it as a fraction, decimal or percentage.
The Probability Scale
Impossible 1/4
Unlikely 1/2
Even chance 3/4
Likely 1
Certain
Theoretical Probability Formula
P(event) = Number of favourable outcomes
Total number of equally likely outcomes
Sample Space
The sample space is the complete list of all possible outcomes in an experiment. Listing the sample space helps you count outcomes accurately.
Flipping a Coin
Sample space = {Heads, Tails}
P(Heads) = 1/2
Rolling a Die
Sample space = {1, 2, 3, 4, 5, 6}
P(even number) = 3/6 = 1/2
Experimental vs Theoretical Probability
Theoretical Probability
What we expect to happen based on equally likely outcomes, calculated without doing the experiment.
For a fair coin: P(H) = 1/2 = 50%
Experimental Probability
What actually happens when you conduct an experiment or trial. Calculated from real results.
If you flip 20 times and get 9 heads: P(H) = 9/20 = 45%
Key Insight: More Trials = Closer to Theoretical
As you do more trials, experimental probability gets closer to theoretical probability. This is called the Law of Large Numbers.
| Trials | Heads | Exp. Prob. | Theory |
|---|---|---|---|
| 10 | 4 | 40% | 50% |
| 50 | 24 | 48% | 50% |
| 1000 | 501 | 50.1% | 50% |
Key Vocabulary
Probability
A number between 0 and 1 that measures how likely an event is to occur. 0 = impossible, 1 = certain.
Sample Space
The complete list of all possible outcomes for an experiment. E.g. rolling a die: {1, 2, 3, 4, 5, 6}.
Theoretical Probability
Probability calculated using mathematical reasoning, based on equally likely outcomes. No experiment needed.
Experimental Probability
Probability found from actual trials. Calculated as: favourable outcomes ÷ total trials.
Worked Examples
A bag contains 3 red, 4 blue and 1 green marble. Find P(blue).
Total marbles: 3 + 4 + 1 = 8
Favourable outcomes (blue): 4
P(blue) = 4/8 = 1/2
A spinner is spun 40 times. It lands on red 14 times. What is the experimental probability of red?
Favourable outcomes: 14 (red)
Total trials: 40
P(red) = 14/40 = 7/20 = 0.35 = 35%
List the sample space for choosing a vowel from the word MATHS.
Letters in MATHS: M, A, T, H, S (5 letters total)
Vowels: A (only one vowel)
P(vowel) = 1/5. Sample space = {M, A, T, H, S}
Knowledge Check
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Key Concepts Summary
- ●Probability is expressed as a fraction, decimal or percentage between 0 and 1.
- ●The sample space lists all possible outcomes of an experiment.
- ●Theoretical probability = favourable outcomes ÷ total equally likely outcomes.
- ●Experimental probability = favourable results ÷ total trials conducted.
- ●With more trials, experimental probability gets closer to theoretical probability.