Comparing theoretical and experimental probabilities

What is experimental probability?

In math, when we deal with probability, we may be asked for the experimental probability of an experiment. What this means is that they're looking for the probability of something happening based off the results of an actual experiment. This is the experimental probability definition.

So for example, if you're asked for the probability of getting heads after flipping a coin 10 times, the experimental probability will be the number of times you got heads after flipping a coin 10 times. Let's say that you got 6 heads out of your 10 throws. Then your experimental probability is 6/10, or 60%.

For theoretical probability, it doesn't require you to actually do the experiment and then look at the results. Instead, the theoretical probability is what you expect to happen in an experiment (the expected probability). This is the theoretical probability definition.

In the case of the coin flips, since there's 2 sides to a coin and there's an equal chance that either side will land when you flip it, the theoretical probability should be $\frac{1}{2}$ or 50%.

Why is there a difference in theoretical and experimental probability? The relationship between the two is that you'll find if you do the experiment enough times, the experimental probability will get closer and closer to the theoretical probability's answer. You can try this out yourself with a coin. You likely won't get exactly 50% for both heads and tails from your first 10 throws, but as you throw a coin 50 times or even 100 times, you'll see the experimental probability's answer getting closer to 50%.

We'll now see how experimental and theoretical probability works with these questions.

Question 1a: Two coins are flipped 20 times to determine the experimental probability of landing on heads versus tails. The results are in the chart below:

What is the experimental probability of both coins landing on heads?

Solution:

We are looking for the experimental probability of both coins landing on heads. Looking at the table in the question, we know that there were 4 out of 20 trials in which both coins landed on heads. So the experimental probability is $\frac{4}{20}$, which equals to $\frac{1}{5}$ (20%) after simplifying the fraction

Question 1b: Calculate the theoretical probability of both coins landing on heads.

Solution:

Now, we are looking for the theoretical probability. First, there are 4 possible outcomes (H,H), (H, T), (T,H), (T, T). 1 out of the 4 possible outcomes has both coins land on heads. So, the theoretical probability is $\frac{1}{4}$ or 25%

Question 1c: Compare the theoretical probability and experimental probability.

From the previous parts, we know that the experimental probability of both coins landing on head equals 20%, while in theory, there should be a 25% chance that both coins lands on head. Therefore, the theoretical probability is higher than the experimental probability.

Question 1d:

What can we do to reduce the difference between the experimental probability and theoretical probability? We can simply continue the experimental by flipping the coin for many more times —say, 20,000 times. When more trials are performed, the difference between experimental probability and theoretical probability will diminish. The experimental probability will gradually get closer to the value of the theoretical probability. In this case, the experimental probability will get closer to 25% as the coins is tossed over more times.

If you're looking for more experimental vs.theoretical probability examples, feel free to try out this question. It'll require you to do some hands-on experimentation!