129i - Explaining Differences in Outcomes

Lesson 129 — Explaining Differences Between Predicted and Observed Outcomes

Strand: Probability | Descriptor: AC9M7P02 | Duration: 45 minutes

Learning Intentions

• To explain why observed results differ from predictions.
• To recognise and refute the gambler’s fallacy.

Success Criteria

I can:

1. Name the possible reasons an observed result differs from a prediction.
2. Explain that individual trials are independent.
3. Refute the gambler’s fallacy with reasoning.
4. Distinguish variation, bias and a wrong model as explanations.

Warmup

(6 minutes — the coin’s memory, whole class vote)

A fair coin has just landed heads six times in a row.

1. What is the probability the next toss is heads?
2. Vote: is tails now “due”?
3. If you saw this happen, what would you actually think?

Collect the votes before answering. Many students — and most adults — say tails is more likely.

Answers: 1. Exactly , unchanged; 2. No — the coin has no memory; 3. Honestly, you might start to suspect the coin is not fair. That is a different thought, and a reasonable one, and today separates the two.

The two responses, distinguished on the board:

• “Tails is due” — the gambler’s fallacy. Wrong.
• “This coin might be biased” — a legitimate hypothesis, testable with more data.

Activities

Activity 1 — Explicit Instruction: Independence (14 min)

The principle:

In a repeated chance experiment with independent trials, the outcome of one trial has no effect on the next.

Why the coin has no memory. The coin is a piece of metal. It does not record its history, and nothing about a previous toss changes its physical properties. Each toss faces the same two equally likely outcomes.

The gambler’s fallacy, named and refuted. The belief that a run of one outcome makes the opposite “due”.

The refutation, in two steps:

1. The physical argument. Nothing carries information from one toss to the next.
2. The counting argument. Before six tosses, HHHHHH is one of equally likely sequences. But we are not asking about the sequence — we are asking about the seventh toss, given the first six have happened. Those six are now fixed. The seventh has two outcomes, still equally likely.

The confusion the fallacy rests on — worth drawing out carefully:

Question	Answer
(seven heads in a row, before any tossing)	— very unlikely
(seventh is heads, given six heads already)	— perfectly ordinary

Both are correct. The first asks about a whole sequence in advance; the second asks about one toss with the past already settled. Confusing them is the fallacy.

Where it does real damage: roulette players betting on red after a run of black; lottery players avoiding recent numbers; people believing they are “due” a win. Casinos rely on this misconception.

We do — true or false, with reasons:

1. After five reds on a roulette wheel, black is more likely.
2. A lottery number that has not appeared for a year is due.
3. If a die has given no sixes in rolls, a six is more likely next.
4. A coin landing heads times in a row is evidence it might be biased.

(Answers: 1. False — the wheel has no memory; 2. False — the balls do not track history; 3. False — still ; 4. True — this is the legitimate hypothesis, not the fallacy. for a fair coin, so scepticism is warranted.)

Q4 is the crucial contrast. Statements 1–3 say the next outcome changes. Statement 4 says the model may be wrong. Only the second kind of reasoning is sound.

Activity 2 — Explaining Differences (14 min)

Pairs. Three possible explanations, applied to real cases.

The three explanations, taught as a diagnostic list:

Explanation	What it means	How to test
Ordinary variation	Chance alone; nothing is wrong	Repeat with many more trials
Bias in the equipment	The die, coin or spinner is not fair	Many more trials; examine the object
A wrong model	Our assumed probabilities were incorrect	Re-examine the assumptions

For each case, decide which explanation is most likely and say how you would test it:

1. A coin tossed times gives heads.
2. A die rolled times gives sixes.
3. A spinner assumed to have gives red over spins; on inspection, the red sectors are visibly wider.
4. A drawing pin lands point-up of the time; the class predicted because “there are two outcomes”.
5. A bag believed to hold red and blue gives red of the time over draws.
6. A coin tossed times gives heads.

Socratic scaffolding:

Prompt	Purpose
Case 2: predicted sixes?	. Observed — a huge excess over trials.
Which explanation?	Bias — chance essentially cannot do this.
Case 3: what was wrong?	Not chance and not a faulty spinner — our model assumed equal sectors when they were not.
Case 4: where did come from?	The two-outcomes fallacy from Lesson 121 — a wrong model, not a strange pin.
Case 5: what should be checked?	Whether the bag actually contains and — count the counters. The model may be wrong.
Cases 1 and 6?	Ordinary variation in both — small runs, modest deviations.

(Answers: 1. variation; 2. bias; 3. wrong model — unequal sectors; 4. wrong model — outcomes were never equally likely; 5. wrong model — check the bag’s contents; 6. variation.)

The diagnostic order to record: with a small run, suspect variation first. With a large run and a persistent gap, suspect bias or a wrong model — and check the model before blaming the equipment, because a wrong assumption is far more common than a loaded die.

Activity 3 — Inquiry: the Casino Argument (9 min)

Pairs, then class discussion.

A gambler has lost spins in a row on red. He says: “Red is bound to come up now — I’ve been losing all night. I’m due.” He doubles his bet.

1. Is his reasoning about probability correct?
2. What is the probability the next spin is red? (Assume on a European wheel.)
3. Does the run of losses change anything at all about the next spin?
4. Is there any legitimate conclusion he could draw from ten losses?
5. Why do casinos display the recent results on a screen beside every roulette table?

Socratic scaffolding:

Prompt	Purpose
Q1: is he right?	No — the classic gambler’s fallacy.
Q2: the probability?	— unchanged, and slightly under half.
Q3: anything at all?	Nothing. The wheel is physically unaffected by its history.
Q4: any legitimate conclusion?	With enough data he could question whether the wheel is biased — but ten spins is far too few, and … which is uncommon but happens constantly across thousands of players.
Q5: why the screen?	Precisely to encourage the fallacy. Displaying history has no informational value for a fair wheel — its purpose is to prompt bets.
Looking back	The mathematics is simple; the psychology is powerful. Knowing the fallacy is named and understood is the defence.

The closing point: every result in this block — the coin settling near , the pooled data beating individual runs, the independence of trials — combines into one practical piece of knowledge: chance has no memory, and only many trials reveal the truth.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. A fair coin lands tails five times in a row. Find .
2. Name the fallacy in “black is due after five reds” and explain why it is wrong.
3. A die rolled times gives sixes. Which explanation is most likely, and how would you test it?
4. Name the three possible explanations for an observed result differing from a prediction.
5. Reasoning. Explain the difference between “the next toss is more likely to be tails” and “this coin might be biased”, and say which can be reasonable.

Answers: 1. ; 2. The gambler’s fallacy — the wheel has no memory and each spin is independent, so previous results cannot change the next; 3. Predicted ; observed over trials is a very large excess — bias is most likely; test with several thousand more rolls and by examining the die; 4. Ordinary variation, bias in the equipment, or a wrong model; 5. The first claims the next outcome has changed, which is false for independent trials. The second questions whether our assumed probability was right, which is a legitimate hypothesis testable with more data. Only the second can be reasonable.

Common Misconceptions

Misconception	How to pre-empt it
The gambler’s fallacy.	Named, refuted two ways, and revisited in the casino inquiry.
Confusing with .	The two-row table — the confusion the fallacy rests on.
Believing any deviation means faulty equipment.	The three-explanation diagnostic; check the model first.
Thinking “might be biased” is the same fallacy.	Case 4 of the true/false set draws the contrast explicitly.
Assuming a wrong model is rare.	Cases 3–5 are all wrong models; loaded dice are far rarer.
Believing recent results carry information.	The roulette screen’s purpose.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A fair die has given no sixes in rolls. Find .

Answer

— unchanged. The die has no memory.

E2 (AMC Junior style). Find with a fair coin. Then find .

Answer

for the sequence in advance; for the fourth toss once the first three are settled. Both are correct answers to different questions.

E3 (Challenge). A spinner assumed fair with sectors gives red of the time over spins. List two possible explanations and say how you would distinguish them.

Answer

Either the spinner is physically biased (unequal sectors, a weighted pointer) or the assumption of four equal sectors was wrong. Distinguish by measuring the sectors — a wrong model is discoverable by inspection, whereas subtle physical bias is not.

E4 (Challenge). Explain why a lottery player avoiding numbers that came up last week gains no advantage.

Answer

Lottery draws are independent — the balls carry no record of previous draws. Every combination remains equally likely, so avoiding recent numbers neither helps nor harms. (It may marginally reduce the chance of sharing a prize, if other players favour those numbers — but that is a fact about people, not about probability.)

E5 (Challenge). A student argues: “Ten heads in a row is so unlikely that the coin must be biased.” Assess the argument carefully.

Answer

for a fair coin — genuinely unlikely, so scepticism is reasonable. But the argument is incomplete: across a class of thirty students each tossing ten times, someone getting all heads is not remarkable. The right response is to test further with hundreds more tosses, not to conclude from one run.

Homework

1. A fair coin lands heads four times in a row. Find and explain.
2. Name and define the gambler’s fallacy, with an example.
3. True or false, with reasons: (a) after three sixes, a six is less likely (b) a coin landing heads times in a row is worth investigating (c) a lottery number not drawn recently is due (d) each spin of a roulette wheel is independent.
4. Name the three possible explanations for an observed result differing from a prediction, with an example of each.
5. For each case, choose the most likely explanation: (a) a coin gives heads over tosses (b) a die gives sixes over rolls (c) a “fair” spinner with visibly unequal sectors gives red instead of the predicted .
6. Explain, using both the physical and the counting argument, why a coin has no memory.
7. Explain the difference between and , giving both values.
8. Reasoning. Why is “this coin might be biased” a scientific statement while “tails is due” is not?
9. Reasoning. Why do casinos display recent results beside roulette tables?
10. Challenge. A friend has lost eight coin tosses in a row and wants to bet double on the ninth, arguing they are due. Write a short reply explaining why the reasoning fails, and what — if anything — eight losses could legitimately suggest.

Answers: Q1 — ; each toss is independent and the coin’s physical properties are unchanged. Q3 — (a) false (b) true — , so scepticism is warranted (c) false (d) true. Q5 — (a) ordinary variation (b) bias — against over rolls is far beyond chance (c) a wrong model: the sectors were never equal. Q7 — and . Q8 — “might be biased” makes a claim about the coin that further data can support or refute; “tails is due” claims that past results change future probabilities, which is false by the definition of independence and cannot be rescued by any evidence. Q9 — to encourage the gambler’s fallacy and prompt further betting; the information has no predictive value for a fair wheel. Q10 — the coin cannot remember; each toss is regardless of history, so doubling the bet does not improve the odds. Eight losses in a row has probability — uncommon but far from impossible, and with many people tossing coins it happens routinely. If they genuinely suspected the coin, the answer would be to test it hundreds more times, not to bet more.