Experimental and Theoretical Probability

Understanding Probability

Probability is a measure of how likely an event is to occur. It ranges from 0 (impossible event) to 1 (certain event).

There are two main types of probability:

• Experimental Probability: Based on actual experiments and real data.
• Theoretical Probability: Based on the possible outcomes in a perfect world.

Experimental Probability

This is calculated by conducting experiments and recording the outcomes. The formula is:

$$\text{Experimental Probability} = \frac{\text{Number of times event occurs}}{\text{Total number of trials}}$$

For example, if you flip a coin 100 times and it lands on heads 55 times, the experimental probability of getting heads is:

Theoretical Probability

This is calculated based on the assumption that all outcomes are equally likely. The formula is:

$$\text{Theoretical Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$

For example, the theoretical probability of rolling a 4 on a fair six-sided die is:

Multiplication of Probabilities

When dealing with independent events, the probability of both events occurring is the product of their individual probabilities.

For example, if the probability of event A is $P(A)$ and event B is $P(B)$, then the probability of both A and B occurring is:

$$P(A \text{ and } B) = P(A) \times P(B)$$

If you roll two dice, the probability of getting a 3 on the first die and a 5 on the second die is: