Theoretical and Experimental Probability

Warm-up

Announce the class lottery: "One name drawn from 24. Amir says his chance is 50-50 — either he wins or he doesn't." Let the room dismantle the claim: outcomes must be EQUALLY LIKELY before counting them, and the 24 names are the equally likely things, not the two storylines.

$P(\text{Amir}) = \frac{1}{24}$. The unit sharpens last year's twin probabilities — theoretical and experimental — into tools for compound events and simulations.

Explore

Sample-space workshop: two coins, then a coin AND a die, then two dice — pairs build complete outcome tables and read probabilities from them ($P(\text{head and even}) = \frac{3}{12} = \frac{1}{4}$; $P(\text{dice sum} = 7) = \frac{6}{36}$).

Then simulation stations: estimate probabilities that resist clean theory — "a family of three children has all girls" simulated with three coin flips × 40 trials versus the theoretical $\frac{1}{8}$; "a 70%-free-throw shooter makes both shots" simulated with a 10-sided spinner (1–7 = make). The comparison of simulated to theoretical (where available) closes the loop from Grade 6.

Formalize

Formalize sample spaces and the counting principle for independent events:

$P(A) = \frac{|A|}{|S|} \qquad P(A \text{ then } B) = P(A) \times P(B) \;\;\text{(independent)}$

The multiplication rule read from the table: the two-dice grid has $6 \times 6$ cells because each of 6 first-outcomes pairs with each of 6 second-outcomes — the product rule IS the table's dimensions. Independence means the first result doesn't tilt the second; drawing WITHOUT replacement breaks it (a preview, flagged honestly).

Practice

Practice: build one sample space table and answer three questions from it; two multiplication-rule computations with independence justified; one simulation design ("describe how to estimate the chance a 4-question true/false quiz is aced by guessing — then compute it: $\frac{1}{16}$"); one spot-the-flaw (an "either it happens or it doesn't = 50%" claim in the wild).

Exit ticket: two spinners, each half red / half blue. $P(\text{both red})$, with the sample space sketched. ($\frac{1}{4}$.)

Exit ticket

Practice: build one sample space table and answer three questions from it; two multiplication-rule computations with independence justified; one simulation design ("describe how to estimate the chance a 4-question true/false quiz is aced by guessing — then compute it: $\frac{1}{16}$"); one spot-the-flaw (an "either it happens or it doesn't = 50%" claim in the wild).

Exit ticket: two spinners, each half red / half blue. $P(\text{both red})$, with the sample space sketched. ($\frac{1}{4}$.)