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Predictions Using Probability

This lesson explains how to use theoretical and experimental probability to predict outcomes of future events. It covers the formulas for each type, works through dice and spinner examples, and shows how predictions become more reliable with more trials.

Introduction

You already know how to predict what might happen next in a story using clues from the text. Probability gives you a different kind of prediction tool, one built on numbers instead of story clues. Instead of guessing based on character traits or foreshadowing, you use theoretical and experimental probability to predict how often an event is likely to happen.

Theoretical Probability: Predicting Before You Try

Theoretical probability tells you what should happen based on all the possible outcomes, without actually running the event. You find it with this formula: \( P = \dfrac{f}{n} \), where f is the number of favorable outcomes and n is the total number of possible outcomes.

Suppose you roll a standard six-sided die. There is one favorable outcome (rolling a 6) out of six possible outcomes, so \( P(6) = \dfrac{1}{6} \). You can use this probability to predict how many times you expect to roll a 6 in a set number of rolls. If you roll the die 60 times, your prediction is \( \dfrac{1}{6} \times 60 = 10 \). This does not mean you will roll exactly ten 6s every time, but it tells you the number you should expect.

Experimental Probability: Predicting From Real Data

Experimental probability comes from actual trials instead of counting all possible outcomes. The formula is \( P = \dfrac{o}{t} \), where o is the number of times the event actually occurred and t is the total number of trials you ran.

Imagine you spin a colored spinner 50 times and record the results shown below.

Frequency of each spinner color across 50 trials.

The spinner landed on red 20 times out of 50 spins, so the experimental probability of red is \( P(red) = \dfrac{20}{50} = 0.4 \). You can use this result to predict future outcomes even without knowing how the spinner is built. If you plan to spin it 200 more times, your prediction is \( 0.4 \times 200 = 80 \) red spins.

Key Terms & Definitions

Theoretical Probability: A prediction based on counting all possible outcomes, found with \( P = \dfrac{f}{n} \).

Experimental Probability: A prediction based on results actually collected from trials, found with \( P = \dfrac{o}{t} \).

Favorable Outcome: An outcome that matches the specific event you are trying to predict.

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

Frequency: The number of times a particular outcome occurs during a set of trials.

Sample Space: The complete list of all possible outcomes for an event.

Prediction: A reasonable estimate of a future outcome, based on a calculated probability.

Using Probability to Predict Future Outcomes

Once you know a probability, either theoretical or experimental, you can predict how many times an event should happen in a larger number of trials. Use \( N = P \times t \), where P is the probability, t is the number of future trials, and N is the predicted number of times the event occurs.

For example, if a bag has 3 red marbles and 7 blue marbles, the theoretical probability of drawing red is \( P(red) = \dfrac{3}{10} \). If you plan to draw a marble, record its color, and put it back 40 times, your prediction is \( \dfrac{3}{10} \times 40 = 12 \) red draws.

Comparing Experimental and Theoretical Probability

Theoretical and experimental probability do not always match exactly, especially with a small number of trials. A coin might land heads 7 out of 10 times even though the theoretical probability is \( \dfrac{1}{2} \). As you collect more trials, experimental probability usually moves closer to the theoretical value. This is why scientists and researchers run experiments many times before trusting their predictions.

Checking your experimental results against theoretical predictions is a useful way to judge whether something unusual is happening, like a weighted die or an unfair spinner. Once you are comfortable with these basic probability predictions, you can build on this skill by working with larger, real-world data sets in Predictions from Data Trends and by applying formal prediction tools in Predictions Using Statistical Models.

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Making Predictions