📚 Practice With Worksheets
Practice sample spaces, compound events, and theoretical 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 compound probability, compare predicted and observed results, and extend your understanding to events with and without replacement.
🛠️ How to Use This Tool
Choose an event such as an exact ordered pair, a pair in either order, a target sum, at least one specified value, or doubles.
Run one or several trials, watch the red and green dice, and use the result cards to compare the selected event’s theoretical probability with the accumulated experimental probability.
Study the 36-outcome sample-space matrix to see which ordered pairs favor the event and where the current roll appears.
🧠 Things to Notice
Each ordered pair is written as (red die, green die), so changing the order may create a different outcome.
Sums near 7 have more favorable ordered pairs than extreme sums such as 2 or 12, so they occur more often.
Experimental probability may vary widely in a small number of trials, but it often moves closer to the theoretical probability as the number of trials increases.
📘 Vocabulary
Compound event: An event made from two or more simple events, such as rolling a red die and a green die.
Sample space: The complete set of possible outcomes for an experiment.
Ordered pair: Two values written in a fixed order. In this tool, the first value is the red die and the second is the green die.
Favorable outcome: An outcome that satisfies the selected event.
Theoretical probability: The probability predicted from all equally likely outcomes in the sample space.
Experimental probability: The probability calculated from the results of actual trials.
Trial: One performance of the probability experiment.
Law of large numbers: The idea that experimental results tend to approach the theoretical probability as the number of trials becomes large.
🎲 Key Probability Concepts
Rolling two six-sided dice creates 36 equally likely ordered outcomes because each of the 6 red-die results can be paired with each of the 6 green-die results.
Theoretical probability compares the number of favorable outcomes with the total number of outcomes. For example, a sum of 7 has 6 favorable ordered pairs out of 36, so its probability is 6/36 = 1/6 ≈ 0.167 ≈ 16.7%.
Experimental probability compares the number of successful trials with the total number of trials actually completed. It may not match the theoretical value exactly, especially during a short experiment.
The sample-space matrix explains why some events are more likely than others: events with more highlighted cells have more favorable outcomes and therefore a greater theoretical probability.
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