Making Predictions Using Probability

You already know how to predict what happens next in a story using clues and evidence. Now you will use that same predicting mindset with numbers. Probability lets you make a mathematical prediction about an outcome, test it with real data, and then revise your thinking if the results surprise you. This builds on the prediction and revision process from Making Predictions Adjusting Understanding, but instead of reading clues in a text, you read clues in data.

Theoretical probability is a prediction based on reasoning alone, before you run any trials. You look at all the possible outcomes and figure out how many of them count as a "favorable" outcome, meaning the one you are interested in.

The formula is \( P(A) = \dfrac{f}{n} \), where f stands for the number of favorable outcomes and n stands for the total number of possible outcomes. For example, if you roll a fair six sided die, the probability of rolling a 4 is \( P(4) = \dfrac{1}{6} \), because there is one favorable outcome out of six total outcomes.

You can use theoretical probability to predict how many times an event should happen over several trials with the formula \( E = P \times n \), where E is the expected count and n is the number of trials. If you flip a coin 40 times, you would predict about \( 0.5 \times 40 = 20 \) heads.

Experimental probability comes from real trials instead of reasoning. You count how many times an event actually occurred and divide by the total number of trials you ran. The formula is \( P_{exp} = \dfrac{r}{n} \), where r is the number of times the event occurred and n is the total number of trials.

Suppose you spin the spinner below 20 times and land on yellow 7 times. Your experimental probability for yellow is \( P_{exp} = \dfrac{7}{20} = 0.35 \), or 35 percent. Compare that to the theoretical probability of landing on yellow, which is \( \dfrac{1}{4} = 0.25 \), or 25 percent, since the spinner has four equal sections.

Notice that the experimental result, 35 percent, does not match the theoretical prediction of 25 percent. This does not mean your math is wrong. Small numbers of trials often produce results that differ from theoretical probability just by chance.

Good predictors treat their first prediction as a starting point, not a final answer. When new data does not match your prediction, you look for a reason before you decide whether to change your thinking. Ask yourself two questions: was the sample size small, or is there a real pattern showing up?

If you only ran 20 trials, a difference between 25 percent and 35 percent is common and does not necessarily mean the spinner is unfair. If you increase your trials to 200 spins and yellow still comes up close to 35 percent of the time, that is a stronger signal. With more trials, experimental probability usually settles closer to theoretical probability, so a large, repeated gap is worth investigating. At that point, you would revise your prediction and consider that the spinner sections might not actually be equal in size.

This same reasoning appears whenever predictions are built from real world data, including more advanced work in Predictions From Statistical Models, where larger and more complex data sets are used to test and adjust predictions.

A bag contains 10 marbles: 4 red, 3 blue, 2 green, and 1 yellow. Predict how many times you would pull a red marble in 50 draws, replacing the marble each time.

Step 1: Find the theoretical probability. \( P(red) = \dfrac{4}{10} = 0.4 \).

Step 2: Predict the expected count. \( E = 0.4 \times 50 = 20 \). You predict about 20 red marbles in 50 draws.

Step 3: Run the trials and record results. Suppose you actually draw red 26 times. The experimental probability is \( \dfrac{26}{50} = 0.52 \).

Step 4: Compare and decide. The gap between 0.4 and 0.52 is fairly large even for 50 trials. A careful predictor would run more trials before deciding whether the bag truly has more red marbles than expected, or whether this result happened by chance.

Theoretical Probability: A prediction based on reasoning about all possible outcomes before any trials happen.

Experimental Probability: A probability calculated from the actual results of real trials.

Favorable Outcome: The specific result you are trying to predict or count.

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

Expected Count: The number of times an event should occur, predicted using theoretical probability and the number of trials.

Sample Size: The total number of trials used to calculate an experimental probability.

Revising a Prediction: Changing your prediction after new data shows a pattern that contradicts your original reasoning.

Strengthen this skill by making a theoretical prediction first, then running an actual experiment to test it. Try flipping a coin, rolling a die, or spinning a spinner at least 20 times and recording every result in a table. Compare your experimental probability to your theoretical probability, and write a sentence explaining whether the difference is likely due to a small sample size or a real pattern.

Related lessons

• Making Predictions Adjusting Understanding