Lesson 2: Theoretical vs. Experimental Probability

Property

Theoretical Probability is what we expect to happen in an ideal situation. It is calculated based on the possible outcomes of an event.
Experimental Probability is what actually happens when we conduct an experiment. It is calculated based on the results of trials or observations.

Examples

• Coin Flip: A fair coin has an equal chance of landing on heads or tails. This represents theoretical probability, because it is based on how the coin is designed. Recording the results after flipping a coin several times represents experimental probability, because it is based on observed outcomes.

• Rolling a Die: A standard six-sided die has an equal chance of landing on each number from 1 to 6. This represents theoretical probability, because it is based on the structure of the die. Rolling the die multiple times and using the results to describe how often a 4 occurs represents experimental probability, because it comes from an experiment.

Property

When the probability of an event is known, or can be determined through analysis where all outcomes are equally likely, the theoretical probability is:

Examples

• The theoretical probability of rolling a 2 on a six-sided die is $\frac{1}{6}$. If you roll it 12 times and get a 2 three times, the experimental probability is $\frac{3}{12}$ or $\frac{1}{4}$.
• A bag has 4 red and 6 blue marbles. The theoretical probability of drawing red is $\frac{4}{10} = \frac{2}{5}$. After drawing and replacing 20 times, you draw red 9 times. The experimental probability is $\frac{9}{20}$.
• A spinner has 4 equal sections. The theoretical probability of landing on 'A' is $\frac{1}{4}$. After 60 spins, it lands on 'A' 12 times. The experimental probability is $\frac{12}{60} = \frac{1}{5}$.

Explanation

Theoretical probability is what should happen based on pure math, like a $\frac{1}{2}$ chance of heads. Experimental probability is what actually happens when you run the experiment, like getting 47 heads in 100 flips.