Expected value and fair games (IB Math AI SL)

Key ideas

• Write out the probability distribution first and check the probabilities add to 1; use this to find an unknown probability.
• E(X) = Σ x P(X = x) is the long-run average per trial, not a value you will get on one go.
• For a game, let X be the player's gain: the prize minus the cost to play. The game is fair if E(X) = 0.
• If X is the prize only, the game is fair when the cost to play equals E(X).
• Say what the answer means in context: a negative expected gain means the player loses money on average.

Worked example

A game costs $3 to play. A fair six-sided die is rolled once: a 6 wins $10, a 4 or a 5 wins $2, and anything else wins nothing. Find the player's expected gain per game, decide whether the game is fair, and find the cost that would make it fair.

1. Gains: 6 gives 10 − 3 = $7 (probability 1/6); 4 or 5 gives 2 − 3 = −$1 (probability 2/6); 1, 2 or 3 gives −$3 (probability 3/6).
2. E(gain) = 7 × 1/6 + (−1) × 2/6 + (−3) × 3/6 = (7 − 2 − 9)/6 = −2/3 ≈ −$0.67.
3. E(gain) ≠ 0, so the game is not fair: the player loses about 67 cents a game on average.
4. Expected prize = 10 × 1/6 + 2 × 2/6 = 14/6 ≈ $2.33, so a cost of $2.33 would make it fair.

Answer: Expected gain −$0.67, so not fair; a fair cost is $2.33.

Exam-style SL AI questions

• SL AI Statistics and Probability questions by topic — worked examples and practice questions for every topic in this unit

Maths IA ideas that use expected value and fair games

• Is a lottery ticket ever worth buying? Expected value and rollovers
• Where should you build your first settlement in Catan?
• Radioactive decay with dice: does chance produce an exponential curve?
• Can you beat people at rock–paper–scissors?
• All SL AI IA ideas

Related in Statistics and Probability

• Chi-squared test — goodness of fit and independence
• Correlation, regression, and Pearson's r
• Normal distribution — CG50 and TI-Nspire keystrokes

Revise Expected value and fair games for the exam

• SL AI revision notes — every unit summarised, with the key results to learn
• SL AI past papers — specimen papers and timed practice papers
• Formulas for Statistics and Probability — on the one-page SL AI formula sheet (free PDF), with the ones not in the IB booklet marked
• SL AI formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

TOK link: Can a single average represent a group fairly?

Stretch yourself: original probability problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S32 · Problem S105 · Problem S33 · Problem S106. All 20 →