Key takeaway: Theoretical probability is the chance of an event worked out by reasoning, as favorable outcomes divided by all outcomes when each outcome is equally likely.
Grade 77.SP.C.7aProbability and Counting
Definition of Theoretical Probability
Theoretical probability tells you what should happen based on the structure of the situation, before anything is tried. A fair spinner with 8 equal sections has no reason to favor any number, so each section gets the same chance. You count, you do not experiment.
The method needs outcomes that are equally likely. A fair die, a fair coin, a well-mixed bag and an evenly divided spinner all qualify. A loaded die, a lopsided spinner or a coin that has been bent do not, so counting would be misleading. The answer may differ from what a handful of actual trials produce, and that gap is normal.
Formula and notation
For an event \(E\) among equally likely outcomes:
\[P(E) = \frac{\text{number of outcomes in } E}{\text{number of outcomes in the sample space}}\]
The result is a fraction, decimal or percent between 0 and 1.
Key facts
- Theoretical probability needs equally likely outcomes, so a fair device or setup.
- It is found without running any trials, only by counting in advance.
- Results from real trials may differ, but come closer with many trials.
- Summing the chances of every single outcome always gives exactly 1.
Theoretical Probability example
A fair spinner has 8 equal sections numbered 1 through 8. Find the probability that it lands on a multiple of 3, and then on a number greater than 5.
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Multiples of 3 between 1 and 8: 3 and 6, so 2 outcomes. \(P = \frac{2}{8} = \frac{1}{4}\).
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Numbers greater than 5: 6, 7 and 8, so 3 outcomes. \(P = \frac{3}{8}\).
The chances are \(\frac{1}{4}\) for a multiple of 3 and \(\frac{3}{8}\) for a number above 5.
Common mistake
Watch out
Isla reasons that the spinner either lands on a multiple of 3 or it doesn’t, so there are 2 outcomes and the chance is \(\frac{1}{2}\).
Why it is wrong
For theoretical probability the cases “is a multiple of 3” and “is not” are not equally likely. The equally likely outcomes are the 8 sections, and only 2 of them are multiples of 3. The correct value is \(\frac{2}{8} = \frac{1}{4}\). Count the equal pieces, not the categories.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Probability and Counting