Theoretical Probability
In short: Theoretical probability predicts an event’s chance by pure reasoning: count the outcomes that fit, then divide by the count of all outcomes that are equally likely.
7th GradeProbability7.SP.C.7a
What is Theoretical Probability?
The theoretical probability of rolling a 6 is one in six, and nobody needs thousands of rolls to know it. It comes from reasoning about the setup itself, assuming every outcome has the same chance.
The formula is \(P(E) = \frac{n(E)}{n(S)}\), where \(n(E)\) counts the outcomes in the event and \(n(S)\) counts every outcome. It only works when the outcomes are equally likely, as with a fair coin, a fair die or a well-mixed set of cards.
How to spot it
- The objects are fair: a standard die, a balanced spinner, a shuffled deck, numbered cards.
- Nobody runs a trial; the answer comes straight from counting possibilities.
- The problem gives the total number of outcomes and asks for a fraction for some of them.
- Words like fair, equally likely or at random appear.
The method
- Confirm that every outcome is equally likely; if not, this formula does not apply.
- Tally every outcome in the sample space; that total becomes the bottom of the fraction.
- Tally the outcomes that satisfy the event; that count goes on top.
- Simplify the fraction and, if asked, convert it to a decimal or percent.
Theoretical Probability: solved example
Problem
A game uses a fair 12-sided die with faces numbered 1 through 12. What is the theoretical probability of rolling a multiple of 5?
Solution
The sample space has 12 equally likely faces. The multiples of 5 in that range are 5 and 10, so there are 2 favorable faces.
\(P = \frac{2}{12} = \frac{1}{6}\). The probability is \(\frac{1}{6}\), about 16.7%.
Try it
Question 1
One card is pulled blindly from a stack numbered 1 through 20. How likely is a prime number?
Show the answer
The primes are 2, 3, 5, 7, 11, 13, 17, 19, which is 8 cards: \(\frac{8}{20} = \frac{2}{5}\).
Question 2
A whole number from 1 to 30 is chosen blindly. How likely is it that the number is not a perfect square?
Show the answer
The squares are 1, 4, 9, 16, 25, so 25 numbers are not squares: \(\frac{25}{30} = \frac{5}{6}\).
Theoretical Probability vs. Experimental Probability
Theoretical probability predicts from the structure of the situation, for example one chance in six for a fair die. Experimental probability reports what actually happened in recorded trials. A die rolled 30 times might show a 6 only 3 times, a value of 10%, even though the prediction is about 16.7%.
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Whole dictionary: A–Z · 7th Grade · Probability