Problem 54 - College Algebra

Question

Describe the difference between theoretical probability and empirical probability.

Step-by-Step Solution

Answer

Theoretical probability is determined through mathematical calculations, where the probability of an event happening is calculated based on the total number of equally likely outcomes. Empirical probability is determined through actual experiments, where the ratio of the number of an event happening is calculated based on the total number of trials. Theoretically calculated probabilities can be different from the experimental probabilities because of the natural variability or randomness.

Step 1: Define Theoretical Probability

Theoretical probability is calculated by dividing the number of ways an event can occur by the total number of equally likely outcomes. It is the ratio of the favorable outcomes to the total number of possible outcomes. Theoretical Probability relies on the mathematical understanding of the problem.

Step 2: Define Empirical Probability

Empirical probability, on the other hand, is based on actual experiments and recording results to determine the ratio of the number event occurring to the number of trials. Empirical probability is calculated as \( P(E) = \frac{Number\:of\:times\:event\:E\:happened}{Total\:number\:of\:trials} \). It is a probability measure which grows more accurate with the increase in a number of experiments.

Step 3: Comparing Theoretical and Empirical Probability

Theoretical Probability is based on the possible chances of something to happen, whereas empirical probability is based on the experiments done by us. Emphasis should be put on the point that, theoretically calculated probabilities may differ from the actual or experimental probabilities because of the natural variability.