Experiments are an important part of mathematical and scientific discovery. We often have a hypothesis of what we theoretically expect to happen, and then we test it with an experiment. Sometimes the theory and experiment match up, sometimes they do not.
Experimental Probability = \frac{\text{frequency of the event}}{\text{total number of trials}}
A die is rolled $60$60 times and the results are recorded in the following table:
Number | Frequency |
---|---|
$1$1 | $10$10 |
$2$2 | $12$12 |
$3$3 | $8$8 |
$4$4 | $10$10 |
$5$5 | $8$8 |
$6$6 | $12$12 |
What is the experimental probability of rolling a $6$6 with this die?
Express your answer in simplest form.
What is the experimental probability of rolling a $3$3 or higher with this die?
Express your answer in simplest form.
What is the experimental probability of rolling a $3$3 or lower with this die? Express your answer in simplified form.
As the previous example demonstrates, theoretical probability is not always reflected in the experimental probability. In theory, there should be a probability of \frac{1}{6} for rolling any particular value on a standard die. However, we can see this isn't the case from the above example.
Some ideas or theories need to be confirmed through experiments before determining if they are true or not. For example, if scientists believe they have found a new vaccine, they need to test it to check if it does indeed work until they actually do the experiments, we won't know for sure. In mathematics, we will often have a hypothesis, test it with a few numbers to check and then prove it mathematically using other known facts.
Let's go back to the example above of testing a vaccine. How many trials do you think the scientists would do before deeming the vaccine safe? One, five, one hundred, one thousand? They would want reliable and consistent results before making a conclusion.
A more trivial example would be determining if a die is fair or biased. Consider the questions below to get started.
Use the animation below to see how theory compares to reality.
In the movie, The Dark Knight, one of the main characters has a coin with a heads on both sides and claims that he "makes his own luck." While we might likely notice if someone had a coin with two heads, there is another trick which is weighting one side of the coin, so it is more likely to land down on that side.
If a coin is indeed fair, the experimental probability of flipping heads and tails should both be about 50%. As a class, see how closely the experimental probability is to the theoretical probability of 50%.
In the two experiments above, we see the idea of convergence. This means that as the number of trials gets very large, the experimental probabilities will begin to get closer to the theoretical probabilities. The idea with convergence is that if we continued to roll the die forever, eventually the experimental probability would stop changing. The value it will settle on would be the theoretical probability.