Section 15.3: Experimental and Theoretical Probability

Property

To conduct experimental trials:
(1) Define the experiment and possible outcomes,
(2) Perform the experiment multiple times under identical conditions,
(3) Record each outcome systematically,
(4) Calculate relative frequency as $\frac{\text{number of times event occurs}}{\text{total number of trials}}$

Examples

Property

Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.
A uniform probability model has relative frequency probabilities that are the same for all outcomes.
Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.
A sample space is the list of all possible outcomes. An event is a subset of the sample space.
Theoretical probability is the number of successful outcomes divided by the total number of outcomes.

Examples

• In a class of 20 students, a student is selected at random. The probability that any specific student, like Sam, is selected is $\frac{1}{20}$. This is a uniform model.
• If you toss a thumbtack 100 times and it lands point-up 45 times and point-down 55 times, your model would assign $P(\text{up}) = 0.45$ and $P(\text{down}) = 0.55$. This is a non-uniform model.
• When rolling a standard six-sided die, the sample space is $\{1, 2, 3, 4, 5, 6\}$. The probability of the event 'rolling an even number' is $\frac{3}{6} = \frac{1}{2}$.

Explanation

A probability model is a complete game plan. It lists every possible outcome (the sample space) and the chance of each one happening. This helps compare what we expect (theoretical) with what actually happens (experimental).

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.