Likelihood of Events: Certain, Uncertain, More and Less Likely
Warm-up
Show a spinner with half red and half blue. Is red more likely, blue more likely, or equally likely? (Equally likely: same area.) Show a spinner with 3/4 red and 1/4 blue. Now which is more likely? (Red.) Why? (More of the spinner is red.) Evidence from the visual grounds the judgment.
Explore
Marble bag experiment: each pair has a bag with a known number of coloured marbles (e.g., 6 red, 2 blue). Predict: if you draw without looking, is red or blue more likely? Why? Draw 20 times, replacing each time. Record results. Compare to prediction. Was the data consistent with the likelihood?
Formalize
Connect to class data: show a recent favourite-colour graph. If one more student joins the class today, which colour is more likely to be their favourite? (The most popular colour.) Is this certain? No: uncertain but more likely. This moves from pure probability to data-informed prediction.
Likelihood of Events: Certain, Uncertain, More and Less Likely
Equally likely vs. not equally likely: (1) Rolling a standard die: each number equally likely? Yes. (2) Drawing a card from a deck with more red than black cards: equally likely? No. (3) Choosing a student from a class with 15 girls and 10 boys: equally likely to choose a girl or boy? No. Work through the reasoning for each.
Practice
Students predict and test 4 probability experiments (spinners, marbles, dice), recording predictions, results, and whether results matched predictions. Exit ticket: name one pair of equally likely events and explain why they are equally likely.
Exit ticket
Students predict and test 4 probability experiments (spinners, marbles, dice), recording predictions, results, and whether results matched predictions. Exit ticket: name one pair of equally likely events and explain why they are equally likely.
Step 1: Draw a long line on the board: IMPOSSIBLE at the left end, CERTAIN at the right, LESS LIKELY / MORE LIKELY as the stretch between.
Step 2: Place events by argument, not vote: "Tomorrow has a morning" → certain (pin it at the far right). "You'll roll a 9 on one die" → impossible (far left — inspect the die: faces are 1–6). "You'll hear a dog bark today" → likely, but not certain (right of middle). "Snow in June here" → unlikely, not impossible (left of middle).
Step 3: For each pin, require the sentence: "___ because ___." The BECAUSE forces the reasoning: what do we know about dice, dogs, and June?
Step 4: The subtle wins to highlight: certain and likely are different ranks (barking is very likely, yet a silent day is possible); impossible and unlikely are different ranks (a 9 CANNOT happen; June snow just mostly doesn't).
Exit ticket: each student writes one event per zone — the inventing is harder than the sorting.
Step 1: Show a bag loaded with 8 blue marbles and 2 red. Ask for predictions with reasons: "blue is MORE LIKELY because there are more blues in there."
Step 2: Draw one marble, show it, tally it, PUT IT BACK (returning it keeps the bag the same for every draw — worth saying out loud). Repeat 20 times as a class, tallying each result.
Step 3: Read the tally: something like blue 16, red 4. The results lean blue — as predicted — but red DID appear. More likely never meant "always."
Step 4: Flip the reasoning direction: hide a new mystery bag, draw-and-return 20 times, and ask students to GUESS the bag's contents from the tally (say, 15 yellow / 5 green → "mostly yellow, maybe a few green"). Reasoning from results back to contents is real inference — the same move scientists make.
The moment: a half-red half-blue spinner lands blue five times running. A student declares red is "due next" — the spinner owes them.
Step 1: Take the claim seriously enough to test it: does the spinner REMEMBER? Examine it — where would the memory live? The arrow doesn't know history; each spin starts fresh from a flick.
Step 2: Reason from the space: red owns half the circle before EVERY spin, including this one. The chances on spin six are exactly what they were on spin one.
Step 3: Address the true intuition hiding underneath: "lots of blues in a row feels weird" — agreed, streaks feel special. But rare-feeling and impossible are different; five blues happens sometimes, and the spinner remains 50/50 throughout.
Step 4: Test with a long run: spin 30 more times as a class, tally. Reds and blues drift toward evenness WITHOUT the spinner ever "catching up" on purpose. The takeaway sentence: "the spinner has no memory."