E(X) as a balance point, and why the expected value is often a number that cannot happen.
Ask where the balance point will be before you reveal it.
Students guess the tallest bar, because “expected” sounds like “most likely”. The beam tips until it settles somewhere that is frequently not a bar at all.
Then give them the 2.5 children example. Nobody has 2.5 children, and the mean is still 2.5. The word is a technical term for a balance point and not a prediction, and the balance beam is the only explanation of that which survives the week.
A table with probabilities summing to 0.95 is not a distribution, and every number computed from it is wrong. Make the sum the first line of working, always, before any multiplying.
Questions that give an unknown k in the table are testing exactly this: the sum to 1 is the equation, and finding k is the first mark.
| 1. E(X) from the table | 0.1 + 0.4 + 0.9 + 1.6 = 3. |
| 2. E(X) for (4 + x)/18 | (5 + 12 + 21) over 18 = 38/18 = 2.111. |
| 3. The fair stake | Expected winnings are 0.3 × 5 = 1.50, so a stake of 1.50 makes the expected gain zero. |
1 markMultiplying each value by its own probability and summing, with the products visible.
1 markChecking or imposing that the probabilities sum to 1.
1 markInterpreting a fair game as expected gain zero, rather than as equal chances.
| They give | What it means |
|---|---|
| 2.5 (Q1) | Averaged the x values and ignored the probabilities. Very common, and it looks plausible on a symmetric-ish table, which is why a deliberately lopsided table is the better teaching example. |
| 4 (Q1) | Gave the most likely value. This is the misreading the balance beam exists to break. |
| 1 (Q1 or Q2) | Summed the probabilities instead. They have done the check and reported it as the answer. |
| 2 (Q2) | Gave the middle value of 1, 2, 3 by symmetry, but the distribution is not symmetric. |
| 6.333 (Q2) | Forgot to divide by 18, or divided only one term. |
| 5 (Q3) | Gave the payout. A fair stake is the expected payout, not the payout. |
"Can E(X) be a value X never takes?" Yes, and that is the point. A fair die has E(X) = 3.5. If this surprises them, it means they were still reading it as a prediction.
"Is a fair game one where I win half the time?" No. Fair means expected gain zero. A game you win one time in ten paying ten times your stake is fair, and feels nothing like it.
"Why do casinos exist?" Because every game has expected gain negative for the player, by design and by a known margin. A two minute digression here does more for retention than another table.
| What is happening | |
|---|---|
| 1 | Guess the balance point. Reveal. Let the gap between guess and answer do the work. |
| 2 | The 2.5 children example and the definition. |
| 3 | The three questions, with the sum-to-1 check written first every time. |
| 4 | A table with an unknown k, then a fair game question. |
| 5 | Where this reappears: the binomial mean next lesson is a special case of exactly this. |
Do not say “the expected value is what you expect to get”. It is circular and it reinforces the misreading.
Do not only use symmetric distributions. They let the wrong method give the right answer, which is the worst possible feedback.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.