LESSON PLAN

Probability: Theoretical and Experimental

Apothem Team

Grade 6 · Data & Probability

LESSON AT A GLANCE

Warm-up

5 min

Explore

15 min

Formalize

10 min

Practice

12 min

Exit ticket

3 min

Warm-up

Hold up a thumbtack: "Heads a coin is fair — but if I toss THIS, what's the chance it lands point-up?" Students realize no symmetry argument helps: a tack isn't a coin; its outcomes aren't equally likely.

The gap this exposes is the unit's engine: THEORETICAL probability (from symmetric counting) versus EXPERIMENTAL probability (from doing it many times). The tack demands the second kind.

Explore

Two-station comparison: (1) The die station: predict $P(\text{six}) = \frac{1}{6}$ theoretically, then roll 60 times and compute the experimental fraction — compare. (2) The tack station: NO theory allowed; toss 50 times, tally point-up, and let the experimental fraction BE the estimate (maybe $\frac{31}{50} \approx 0.62$ ).

Pool the class's tack data and watch the estimate stabilize as trials mount. Close with the definitions written from experience: theoretical = favourable ÷ total (when outcomes are equally likely, argued from symmetry); experimental = observed ÷ trials (always available, sharpens with more data).

Formalize

Formalize both probabilities and their relationship:

$P_{\text{theo}} = \frac{\text{favourable outcomes}}{\text{equally likely outcomes}} \qquad P_{\text{exp}} = \frac{\text{times it happened}}{\text{trials}}$

The bridge law, stated honestly: as trials grow, experimental probability tends to drift toward the theoretical value (when one exists). Disagreement in small samples is normal wobble; persistent disagreement in large samples indicts the assumptions (loaded die, biased spinner) — that's not failure, that's DETECTION.

Practice

Practice: compute theoretical probabilities for die, spinner, and card draws; compute experimental probabilities from supplied tally data; one compare-and-judge item ("the spinner shows $P(\text{red}) = \frac{1}{4}$ but 100 spins gave 41 reds — verdict?"); one design task (invent an experiment for a probability with no theory, like a dropped cup landing upright).

Exit ticket: theoretical and experimental probability of an event differ after 10 trials. Give two honest explanations. (Wobble; or wrong theoretical model.)

Exit ticket

Exit ticket: theoretical and experimental probability of an event differ after 10 trials. Give two honest explanations. (Wobble; or wrong theoretical model.)

TIP "Expected" is the most misread word in the unit: expecting 10 sixes in 60 rolls never promises 10. Replace "should get" with "will get NEAR, usually" in all class talk.

WORKED EXAMPLES

Example 1 — Full workup: the number-cube experiment

The task: assess $P(\text{rolling an even number})$ both ways.

Step 1: Theoretical: even faces are 2, 4, 6 → three of six equally likely faces (symmetry argued: uniform cube) → $P = \frac{3}{6} = \frac{1}{2}$ .

Step 2: Experimental: 40 rolls give evens 23 times → $P_{\text{exp}} = \frac{23}{40} = 0.575$ .

Step 3: Compare: 0.575 vs 0.5 — a gap of 0.075 on a small sample. Verdict: consistent with a fair die; no alarm.

Step 4: What WOULD raise the alarm: pooled class data (600 rolls) still showing 0.57+ — a persistent lean in a large sample. Sample size decides how seriously to take a gap; the number alone decides nothing.

Example 2 — Probability from data alone: the cafeteria line

The question: what's the probability a randomly chosen student buys milk at lunch? No symmetry exists — students aren't dice. Data must speak.

Step 1: Collect: over one week, 480 lunches served, 132 included milk.

Step 2: Experimental probability: $\frac{132}{480} = 0.275$ — about 28%.

Step 3: Use it to predict: next week expects roughly $0.275 \times 500 = 137$ or 138 milks for 500 lunches — an ESTIMATE for ordering, not a promise.

Step 4: Interrogate the data's reach: does one week represent all weeks? (Pizza day skews milk? A cold snap?) Experimental probabilities inherit every bias of their sample — the caveat is part of the answer, and saying it is a mark of statistical maturity, not weakness.

Example 3 — The suspicious spinner: wobble or rigged?

The claim: a spinner is labelled half red / half blue. Two groups tested it. Group 1: 20 spins, 13 red. Group 2: 200 spins, 129 red.

Step 1: Compute both experimental probabilities: $\frac{13}{20} = 0.65$ and $\frac{129}{200} = 0.645$ .

Step 2: Judge Group 1 alone: 13/20 on a fair half-half spinner is unusual-ish but comfortably within small-sample wobble. Alone, insufficient evidence.

Step 3: Judge Group 2: 200 spins holding at 0.645 is a different animal — large samples shouldn't lean this far this persistently if the spinner were truly half-half.

Step 4: Verdict and action: suspect the spinner (sticky pivot? unequal regions painted equally?) — and INSPECT it, which reveals the red region is actually noticeably larger. The statistics flagged it; the physical check confirmed it. That two-step — data raises suspicion, inspection settles it — is the scientific method in one classroom prop.

MATERIALS

•Dice and thumbtacks

•Spinners

•Tally sheets

•Class data pooling chart

•Practice set (PDF)

WATCH FOR

!Experimental results expected to match theory exactly; a 14/60 six-count called "wrong." Wobble is the norm — pooling shows the drift toward theory.

!Theoretical probability computed for non-symmetric situations (the tack given "1/2 — up or down"). Equal likelihood must be ARGUED, not assumed.

!Past outcomes believed to influence the next trial ("tails is due"). Independence: the die has no memory.