Investigation: Theoretical probability
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}}
Worked example
question 1
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.
Experiments
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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.
Experimental probabilities and number of trials
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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.
Experiment 1
A more trivial example would be determining if a die is fair or biased. Consider the questions below to get started.
- What is the theoretical probability of rolling a 5 on a standard die as a fraction, decimal and percent?
- If you roll a standard die 60 times, how many times would you expect to roll a 2?
- If you roll a die 1500 times, how many times would you expect to roll a 3?
Use the animation below to see how theory compares to reality.
- Roll the die 6 times by clicking the "Roll Once" button. Do you get one of each value? If not, what is the relative frequency (experimental probability) for each value?
- Roll the die another 4 times by clicking the "Roll Once" button and then use the "roll \times 10" button to get to 60 rolls. Do you get 10 of each value? If not, what is the relative frequency (experimental probability) for each value?
- Use the "Animation" or "roll \times 10" buttons to see what happens to the experimental probabilities as the number of trials increases.
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Experiment 2
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%.
- With a partner or independently, flip a coin 20 times making a tally of how many times it lands on head and tails.
- Find the experimental probability of heads and tails.
- Combine the results for the whole class to find the total frequency of heads, tails and number of trials.
- Find the experimental probability of heads and tails.
Convergence
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.
Discussion questions
- Fill in the blanks in the following sentence. As the number of ______ gets very large, the experimental probability approaches the ______.
- Can you ever be 100% confident in a hypothesis? Explain why or why not? Are there some scenarios where you can be and other where you can't be?
- How might the idea of more trials being more accurate fit into other scenarios?
- There is a balance between doing trials and the cost of the trials. Let's go back to the first experiment of rolling a die. At what point might we say, "we're confident that this die is fair" and stop rolling? Try doing the animation a couple of times and pause it (bottom left corner) when you think it is good enough.
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