When we look at how chances change with different sample sizes, it's important to understand two key ideas: theoretical probability and experimental probability.

Theoretical Probability

Theoretical probability is a simple idea. It's the chance of a specific outcome happening compared to all possible outcomes. This type of probability stays the same no matter how many times we try.

For example, if you roll a fair six-sided die, the chance of rolling a 3 is:

$P(3) = \frac{1}{6}$

This means that no matter if you roll the die just once or a thousand times, the chance of getting a 3 is always 1 out of 6.

Experimental Probability

Now, let’s talk about experimental probability. This type of probability comes from doing tests and looking at the results. It can change a lot based on how many times you do the test. The formula for figuring out experimental probability is:

$P(E) = \frac{\text{Number of successful outcomes}}{\text{Total number of trials}}$

As you try more and more times, the experimental probability usually gets closer to the theoretical probability. This idea is called the Law of Large Numbers.

Impact of Sample Size

Let’s look at an example to make this clearer:

• Small Sample Size: If you flip a coin 10 times and get 7 heads, the experimental probability of getting heads would be:

$P(\text{Heads}) = \frac{7}{10} = 0.7$

• Large Sample Size: If you flip that same coin 1000 times and get 520 heads, the experimental probability would be:

$P(\text{Heads}) = \frac{520}{1000} = 0.52$

Here, when the number of flips went from 10 to 1000, the experimental probability of getting heads got closer to the theoretical probability of 0.5.

Conclusion

To sum it up, theoretical probability doesn’t change no matter how many times you test it. However, experimental probability can jump around with the sample size. Usually, as you try more and more times, it levels out and gets close to the theoretical value. This shows why it’s important to use a good number of trials when you’re testing probabilities.