Updating Predictions With New Data

A prediction about probability is not final the moment you make it. Once you run an experiment and collect real data, you can check your prediction and change it if the data tells a different story. This lesson focuses on revising probability predictions after collecting new data from experiments like coin flips, spinner spins, and dice rolls.

Before running an experiment, you can often make a prediction using theoretical probability. Theoretical probability tells you what should happen based on all the possible outcomes, assuming each outcome is equally likely.

The formula for theoretical probability is \( P_{theory} = \dfrac{a}{b} \), where \(a\) is the number of favorable outcomes and \(b\) is the total number of possible outcomes. For a coin flip, there is 1 favorable outcome (heads) out of 2 possible outcomes, so \( P_{theory} = \dfrac{1}{2} \), or 0.5.

An experiment lets you test a prediction in real life. Each flip, spin, or roll is called a trial. As you run trials, you record what actually happens in a frequency table. A frequency table lists each possible outcome and how many times it occurred.

Once you have trial data, you can calculate experimental probability using \( P_{exp} = \dfrac{f}{n} \), where \(f\) is the number of times the event happened and \(n\) is the total number of trials.

Suppose you predict that flipping a coin will land on heads about half the time, since \( P_{theory} = \dfrac{1}{2} = 0.5 \). Now you run the experiment.

After 10 flips, you record 7 heads. The experimental probability is \( P_{exp} = \dfrac{7}{10} = 0.7 \). This is higher than your prediction of 0.5, but 10 trials is a small amount of data, so it is too early to change your original prediction.

You keep flipping. After 100 flips, you record 46 heads. Now \( P_{exp} = \dfrac{46}{100} = 0.46 \). This result is much closer to the theoretical prediction of 0.5. With this larger set of data, you can update your prediction: the coin still behaves like a fair coin, and the early result of 7 out of 10 was just a small-sample coincidence.

A small number of trials can produce results that look surprising just by chance. As the number of trials grows, the experimental probability usually settles closer to the theoretical probability. This is why collecting more data helps you make a more reliable prediction instead of relying on just a few results.

The same idea applies to spinners, dice, and marbles pulled from a bag. If a spinner has 4 equal sections and you predict landing on the red section about \( \dfrac{1}{4} \) of the time, you can test this by spinning many times and comparing your results. If your prediction was based on limited data, running more trials gives you a chance to revise it and get closer to an accurate estimate, similar to the process described in Revising Conclusions From New Evidence.

This same skill of adjusting an estimate as more information arrives also applies to larger sets of data, which you can explore further in Updating Statistical Models With New Data.

Trial: One instance of running an experiment, such as one coin flip or one spin.

Theoretical probability: The probability calculated from all possible equally likely outcomes, using \( P_{theory} = \dfrac{a}{b} \).

Experimental probability: The probability calculated from actual trial results, using \( P_{exp} = \dfrac{f}{n} \).

Frequency table: A table that records how many times each outcome occurred during an experiment.

Prediction: An estimate of what is likely to happen, which can be revised as new data is collected.

Try rolling a die 20 times and recording the results in a frequency table, then calculate the experimental probability of rolling a 6 and compare it to the theoretical probability of \( \dfrac{1}{6} \). Repeat the experiment for 60 rolls and see whether your experimental probability moves closer to \( \dfrac{1}{6} \).

Related lessons

• Modifying Views Through New Information