Understanding the difference between theoretical and experimental probability is really interesting, and we see it in our daily lives all the time.

Theoretical Probability is what we think should happen in a perfect situation. We figure it out by looking at all the possible outcomes. For example, when you flip a fair coin, there are two outcomes: heads or tails. So, the theoretical probability of getting heads is:

$$P(\text{Heads}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{2}$$

This means there’s a 50% chance of getting heads.

Experimental Probability, on the other hand, is what happens when you actually do an experiment. If you flip that coin 100 times and get heads 45 times, the experimental probability of getting heads would be:

$$P(\text{Heads}) = \frac{45}{100} = 0.45$$

This shows that the two probabilities can be different. This difference can reveal randomness or bias in real-life situations.

From my experience, doing experiments really helps you understand these ideas better. You might think that the results will be close to the theoretical expectations when you do an experiment many times. But it can be surprising to see how often they don’t match up. It’s a great way to learn about the surprises of chance!