📚 Practice With Worksheets

Practice compound and experimental probability with the resources below. The 🛠️ icon identifies a companion worksheet designed specifically for use with the interactive tool above.

▶️ Learn With Videos

Review theoretical and experimental probability, compound events, and probability with and without replacement.

🛠️ How to Use This Tool

Select Run 1 Trial, Run 5 Trials, Run 10 Trials, or Run 50 Trials. Run 1 and Run 5 animate each coin flip and spinner result.

Read each compound outcome as a pair. For example, HR means heads and red, while TY means tails and yellow.

Use Hide Table to enlarge the coin and spinner for classroom projection. Show the table again to compare counts, experimental probabilities, theoretical probabilities, and differences.

🧠 Things to Notice

A fair coin has 2 possible outcomes and the spinner has 4 equal sections, creating 8 equally likely compound outcomes.

Small samples can look uneven. One outcome may appear several times while another may not appear at all, even when all outcomes are equally likely.

As more trials are conducted, the experimental probabilities often move closer to the theoretical probability of \(0.125\) for each compound outcome.

📘 Vocabulary

Compound Event: An event made by combining two or more simple events, such as flipping a coin and spinning a spinner.

Outcome: One possible result of an experiment. In this simulation, examples include HR, HB, and TG.

Sample Space: The set of all possible outcomes of an experiment.

Theoretical Probability: The probability predicted by analyzing all possible outcomes.

Experimental Probability: The probability calculated from the results of actual trials or a simulation.

Relative Frequency: The fraction, decimal, or percent of trials in which an outcome occurs.

Independent Events: Events in which the result of one event does not affect the probability of the other.

Trial: One repetition of a probability experiment.

Law of Large Numbers: The principle that experimental probabilities tend to approach theoretical probabilities as the number of trials increases.

📐 Compound Probability Concepts

The coin has 2 outcomes and the spinner has 4 outcomes. Multiplying gives 8 possible compound outcomes.

\(2\times4=8\text{ possible outcomes}\)

Because the coin and spinner are fair and independent, each specific combination has a theoretical probability of one-eighth.

\(P(\text{HR})=P(\text{heads})\times P(\text{red})=\frac{1}{2}\times\frac{1}{4}=\frac{1}{8}=0.125\)

Experimental probability is based on observed results. For example, if HR occurs 14 times in 100 trials, its experimental probability is 0.14.

\(\text{Experimental }P(\text{HR})=\frac{14}{100}=0.14\)

Theoretical probability stays fixed, but experimental probability changes as more trials are added. According to the law of large numbers, the experimental value often gets closer to the theoretical value over many trials.

\(\text{As trials increase: experimental probability}\rightarrow\text{theoretical probability}\)

Getting closer does not mean the experimental probability must equal the theoretical probability exactly. Random variation can remain even after many trials.

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