Theoretical Probability Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Theoretical Probability.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

Theoretical probability is the expected probability of an event calculated by mathematical reasoning about equally likely outcomes, without conducting experiments. It equals the number of favorable outcomes divided by the total number of possible outcomes.

For a fair coin, you KNOW heads is \frac{1}{2} without flipping. You calculate based on logic: 1 favorable outcome (heads) out of 2 possible outcomes. That's theoretical - it's what SHOULD happen.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Theoretical probability is calculated by counting favorable outcomes and dividing by total equally-likely outcomes, without running any actual experiment.

Common stuck point: Theoretical probability assumes all outcomes are equally likely. If the outcomes are not equally likely (e.g., a weighted die), this formula does not apply.

Sense of Study hint: First, list all possible outcomes (the sample space) and verify they are equally likely. Then count the outcomes that satisfy the event of interest (favorable outcomes). Finally, divide: P(event) = favorable / total.

Worked Examples

Example 1

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Two fair coins are tossed. Find the theoretical probability of getting exactly one head.

Solution

  1. 1
    Step 1: List all outcomes: {HH, HT, TH, TT} — 4 equally likely outcomes.
  2. 2
    Step 2: Outcomes with exactly one head: {HT, TH} — 2 favourable outcomes.
  3. 3
    Step 3: P(\text{exactly one head}) = \frac{2}{4} = \frac{1}{2}.

Answer

\frac{1}{2}
Theoretical probability uses the sample space of all equally likely outcomes. By listing every possibility, we can count favourable outcomes systematically.

Example 2

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A fair die and a fair coin are used together. What is the probability of rolling a 3 AND getting heads?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

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A bag has 4 red and 6 blue balls. One ball is drawn, replaced, and a second is drawn. What is the probability both are red?

Example 2

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Two fair six-sided dice are rolled. What is the theoretical probability that the sum is 7?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

probability basicrelative frequency