Probability as Expectation Examples in Math

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

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

Concept Recap

Probability can be interpreted as the long-run relative frequency of an event over infinitely many identical trials of a random experiment.

P(\text{heads}) = 0.5 means if you flip many times, about half will be heads.

Read the full concept explanation →

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: Probability is a prediction about frequency, not a guarantee about any single trial.

Common stuck point: Individual outcomes can deviate wildly from probability—that's normal.

Sense of Study hint: Try multiplying the probability by the number of trials to get the expected count. That count is an average, not a guarantee.

Worked Examples

Example 1

easy
A basketball player makes free throws with probability 0.75. In 200 free throws, how many do we expect her to make?

Solution

  1. 1
    Expected count formula: E = n \times P
  2. 2
    Substitute: E = 200 \times 0.75 = 150
  3. 3
    Interpretation: on average, she will make 150 of 200 free throws
  4. 4
    Note: this is the long-run average — any single game of 200 shots might yield slightly more or fewer

Answer

Expected makes = 200 \times 0.75 = 150 free throws.
Expected count = n \times P gives the average number of successes in n trials with probability P. This is the long-run mean of repeated experiments, not a guaranteed exact count for any single trial.

Example 2

medium
A game has three outcomes: win \10 (prob 0.2), break even \0 (prob 0.5), lose \$5 (prob 0.3). Calculate the expected value and interpret what it means for 1000 games.

Practice Problems

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

Example 1

easy
A school expects 15% of students to be absent on any given day. If there are 300 students, how many absences are expected?

Example 2

hard
An insurance company charges \200/year for a policy. It pays \10,000 if the insured event occurs (probability 0.01) and \$0 otherwise. Calculate the insurance company's expected profit per policy.
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Background Knowledge

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

probability