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Theoretical Probability and the Probability Scale

Updated July 2026

This section covers the calculation of theoretical probability and its representation on a 0 to 1 scale. Students will learn to relate relative expected frequencies to outcomes using fractions, decimals, and percentages, while applying these principles to practical examples such as rolling a fair six sided die.

Core concept

Theoretical probability is the ratio of favourable outcomes to total possible outcomes in a sample space, expressed by the formula probability=number of favourable outcomesnumber of possible outcomesprobability = \frac{number\ of\ favourable\ outcomes}{number\ of\ possible\ outcomes}.

Defining Theoretical Probability

The relative expected frequency can be used to calculate the theoretical probability of an event. This value represents how likely an event is to occur based on a perfectly fair or idealised model. To find the probability, you compare the specific outcomes you are interested in (the favourable outcomes) to the total number of things that could happen (the possible outcomes).

probability=number of favourable outcomesnumber of possible outcomesprobability = \frac{number\ of\ favourable\ outcomes}{number\ of\ possible\ outcomes}

Probabilities are numeric values that can be represented in three equivalent formats: as fractions, as decimals, or as percentages. For instance, an even chance can be written as 12\frac{1}{2}, 0.50.5, or 50%50\%.

The Probability Scale

Probabilities are always shown on a scale that ranges from 0 to 1, or 0%0\% to 100%100\%. A probability of 0 indicates that an event is impossible, while a probability of 1 indicates that an event is certain to happen. All other probabilities fall between these two extremes.

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Worked Example: Rolling a Fair Six Sided Die

When a fair six sided die is rolled, there are 6 equally likely outcomes: 1, 2, 3, 4, 5, and 6. We can use the probability formula to determine the likelihood of various events.

Landing on the number 3

There is only one face with the number 3.

P(land on 3)=number of ways the die can land on 3number of possible outcomes=16P(\text{land on 3}) = \frac{\text{number of ways the die can land on 3}}{\text{number of possible outcomes}} = \frac{1}{6}

Landing on an even number

The even numbers available on the die are 2, 4, and 6. This gives 3 favourable outcomes.

P(land on an even number)=number of ways the die can land on an even numbernumber of possible outcomes=36=12P(\text{land on an even number}) = \frac{\text{number of ways the die can land on an even number}}{\text{number of possible outcomes}} = \frac{3}{6} = \frac{1}{2}

Landing on a prime number

A prime number is a number with exactly two factors: 1 and itself. On a die, the prime numbers are 2, 3, and 5. There are 3 favourable outcomes.

P(land on a prime)=number of ways the die can land on a prime numbernumber of possible outcomes=36=12P(\text{land on a prime}) = \frac{\text{number of ways the die can land on a prime number}}{\text{number of possible outcomes}} = \frac{3}{6} = \frac{1}{2}

Landing on a square number

A square number is the result of multiplying an integer by itself. On a die, the square numbers are 1 (since 1×1=11 \times 1 = 1) and 4 (since 2×2=42 \times 2 = 4). There are 2 favourable outcomes.

P(land on a square)=number of ways the die can land on a square numbernumber of possible outcomes=26=13P(\text{land on a square}) = \frac{\text{number of ways the die can land on a square number}}{\text{number of possible outcomes}} = \frac{2}{6} = \frac{1}{3}

Key takeaways

  • Probabilities must always be within the range of 0 to 1 inclusive.
  • The theoretical probability is the ratio of favourable outcomes to total possible outcomes.
  • On a standard die, prime numbers are 2, 3, and 5, while square numbers are 1 and 4.
  • Probabilities can be converted between fractions, decimals, and percentages without changing their value.
Tips

When solving die or coin problems, explicitly list your sample space (all possible outcomes) first. This prevents errors in identifying the denominator of your probability fraction.

Cautions

Be careful when identifying prime and square numbers. 1 is a square number but not a prime number. 2 is the only even prime number. Misidentifying these is a common cause of lost marks.

Insight

The 0 to 1 scale is not just for simple dice rolls; it provides a mathematical framework for all uncertainty in science and engineering. As you move into more complex statistics, you will see that the sum of the probabilities of all possible mutually exclusive outcomes always equals 1.

Frequently asked questions

What does a probability of 0.5 represent?

A probability of 0.5, or 50%50\%, represents an even chance, meaning the event is just as likely to happen as it is not to happen.

How do I calculate probability if there are no favourable outcomes?

If there are no favourable outcomes, the numerator is 0, making the probability 0total outcomes\frac{0}{\text{total outcomes}}, which equals 0 (impossible).

Is the number 1 considered a prime number in probability questions?

No, 1 is not a prime number because a prime number must have exactly two distinct factors. On a die, the primes are 2, 3, and 5.

Can I leave a probability as an unsimplified fraction?

While technically correct, it is standard practice in the ESAT to simplify fractions, such as reducing 36\frac{3}{6} to 12\frac{1}{2}, to ensure clarity.

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