Theoretical probability and experimental probability are two ways to evaluate the likelihood of an event occurring, but they are derived from different approaches.
Theoretical Probability
Theoretical probability is based on the principle of equally likely outcomes. It is calculated using a mathematical formula where the probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes. The formula is:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]
Example: Consider rolling a fair six-sided die. The theoretical probability of rolling a 4 is calculated as follows:
- Favorable outcomes: 1 (just the number 4).
- Total possible outcomes: 6 (the numbers 1 through 6).
Thus, the theoretical probability of rolling a 4 is:
\[ P(4) = \frac{1}{6} \approx 0.1667 \]
Experimental Probability
Experimental probability, on the other hand, is based on actual experiments or observations. It is calculated by performing an experiment multiple times, recording the outcomes, and then using that data to find the probability. The formula is:
\[ P(E) = \frac{\text{Number of times event occurs}}{\text{Total number of trials}} \]
Example: If we decide to roll the same six-sided die 60 times and we observe that the number 4 comes up 12 times, the experimental probability of rolling a 4 would be calculated as:
- Number of times 4 occurred: 12
- Total trials: 60
Thus, the experimental probability of rolling a 4 is:
\[ P(4) = \frac{12}{60} = \frac{1}{5} = 0.2 \]
Summary of Differences
- Basis: Theoretical probability is calculated using known possible outcomes, while experimental probability is based on actual outcomes from experiments.
- Formula: Theoretical probability uses a simple fraction of favorable to total outcomes. Experimental probability uses observed data from trials.
- Results: Theoretical probability gives a consistent result based on mathematical logic, whereas experimental probability may vary depending on the number of trials and randomness.
In this example, the theoretical probability of rolling a 4 is approximately 0.1667, while the experimental probability from the trial (if 4 was rolled 12 times out of 60) is 0.2. This illustrates how the two probabilities can differ based on circumstances.