Probability Experiments
Warm-up
Bag with 4 red and 6 blue marbles. What is the theoretical probability of drawing red? (4/10 = 2/5.) Let us test: draw 20 times (replace each time). Record tallies. Calculate experimental probability. Compare to theoretical. Expect values near 2/5 but not exactly.
Explore
Probability experiment design: each group designs their own experiment (spinner, dice, or marble bag), predicts outcomes as fractions, runs 30 trials, records tallies, and calculates experimental probability fractions. Compare theoretical vs. experimental for each group.
Formalize
Lahal/hand game connection: in a guessing game, the probability of guessing correctly on any single guess is 1/2 (the marked item is in one of two hands). Over 10 guesses, expected correct: 10 x 1/2 = 5. In a real game, score might be 3/10 or 7/10 due to chance. What strategies might affect this? (Observing opponent patterns, but randomizing one's own hiding.)
Probability Experiments
Probability scale: place events on a scale from 0 (impossible) to 1 (certain). Rolling a 7 on a standard die: 0/6 = 0. Rolling an even number: 3/6 = 1/2. Rolling any number 1-6: 6/6 = 1. Students see that the fraction 0, 1/2, and 1 correspond to impossible, equally likely, and certain.
Practice
Students run a 40-trial experiment, record with tallies, convert to fractions, and compare to theoretical probability. Play a hand game variant. Exit ticket: a bag has 5 red, 3 blue, 2 green. What is the theoretical probability of drawing green?
Exit ticket
Students run a 40-trial experiment, record with tallies, convert to fractions, and compare to theoretical probability. Play a hand game variant. Exit ticket: a bag has 5 red, 3 blue, 2 green. What is the theoretical probability of drawing green?
The question: flip two coins — is "one head one tail" more, less, or equally likely than "two heads"?
Step 1: The instinct to correct: "three things can happen (HH, TT, mixed) so each is 1 out of 3." Log it.
Step 2: List outcomes properly by DISTINGUISHING the coins (a penny and a dime): penny-H dime-H; penny-H dime-T; penny-T dime-H; penny-T dime-T. FOUR outcomes, all equally likely.
Step 3: Count the target: "mixed" owns two of the four paths (HT and TH); "two heads" owns one. Mixed is TWICE as likely.
Step 4: Test with 40 real double-flips, tallying three bins: HH, TT, mixed. The mixed bin visibly doubles the others.
Step 5: The principle, named: outcomes must be counted at the level where they're EQUALLY LIKELY — and hiding the coins' identities merged two genuine paths into one bin. "How many ways can it happen?" beats "how many labels can I name?" — the same lesson the two-dice sums taught, now in its purest form.
The setup: names go in a jar for a prize draw. Ana's name is on 1 slip. Ben, who won a bonus, has 3 slips. Ten slips total.
Step 1: Compute each chance as a fraction of the slips: Ana 1/10, Ben 3/10. Ben is three times as likely — by design. Is that UNFAIR? Careful: it's unequal, and it's intentional (a bonus). "Fair" can mean equal-chance OR earned-advantage; the math describes, the class debates.
Step 2: The follow-up computations: chance the winner is NEITHER Ana nor Ben: the other 6 slips → 6/10. Chance it's a girl, if girls hold 5 slips → 5/10 = 1/2.
Step 3: Simulate 30 draws (replace the slip each time) and tally: Ben wins roughly 3 in 10 draws — near his fraction, wobble included.
Step 4: Redesign challenge: adjust slip counts so Ben's chance is exactly 1/4 while ten slips remain… impossible with whole slips (2.5)! Change the total to 12: Ben 3/12 = 1/4 ✓. Designing to a target probability is the inverse skill — and it's engineering.
The experiment: a spinner is one-third red, two-thirds blue. In 30 spins a group records red 14, blue 16 — but a third of 30 is 10. "Our spinner is broken!"
Step 1: State the expectation and its true meaning: ABOUT 10 reds — the maths says the long-run fraction, never a promise for any particular 30.
Step 2: Gather the room's data: other groups got red 9, 11, 8, 13, 12. Fourteen sits at the high edge of the pack — unusual-ish, not alien.
Step 3: Do the decisive thing: spin MORE. Pool all groups: 200 spins, red 67 — that's 33.5%, hugging one-third. The wobble that dominated one group's 30 spins dissolved in the crowd's 200.
Step 4: Draw the two lessons in students' words: (1) small samples wobble hard, and a wobble isn't a scandal; (2) the cure for suspicious data is MORE data, not a louder opinion.
Exit: "You flip 10 heads in a row with a fair coin. What's the chance of heads on flip 11 — and what would it take to genuinely suspect the coin?" (½; and something like heads dominating across hundreds of flips.)