Theoretical Probability
Theoretical probability, also known as classical probability, determines the likelihood of an event occurring based on mathematical reasoning rather than conducting actual experiments. It is calculated by dividing the number of outcomes favorable to an event by the total number of equally likely possible outcomes in the sample space.
Key Concepts
It relies on the assumption that all outcomes of a random experiment are equally likely, meaning no single outcome has a higher chance of occurring than another.
The value of theoretical probability always lies between 0 and 1, where 0 represents an impossible event and 1 represents a sure event.
The sum of the probabilities of all the elementary events of an experiment is always equal to 1.
Formula & Equation
P(E) represents the theoretical probability of an event E.
Common Misconceptions
Myth: Believing that theoretical probability guarantees the exact outcome of an experiment if performed a small number of times, such as expecting exactly 5 heads in 10 coin tosses.
Fact: Theoretical probability predicts the long-term average of outcomes. In a small number of trials, actual empirical results may differ, but they tend to get closer to the theoretical probability as the number of trials increases.
Real World Applications
Tossing a fair coin: The theoretical probability of getting a head is 1/2 because there is 1 favorable outcome (head) out of 2 equally likely possible outcomes (head, tail).
Rolling a standard six-sided die: The theoretical probability of rolling an even number is 3/6 (or 1/2) since there are 3 favorable outcomes (2, 4, 6) out of 6 total possible outcomes.
Frequently Asked Questions
What is the difference between theoretical and empirical probability?
Theoretical probability is based on mathematical reasoning and equally likely outcomes, whereas empirical probability is based on the actual results of conducting an experiment.
Can the theoretical probability of an event be greater than 1?
No, the theoretical probability of any event is always a fraction or decimal between 0 and 1, inclusive.