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Ratios and Data 6-8 / Week 12 / Tuesday
2/6
Week 12 · Probability and the Data Fair

Tuesday

Predict, then spin
// Chance from 0 to 1, fair spinners, and your own data report
⏱ about 20 min

Tuesday: Predict, Then Spin

Rocket grips the paper clip pointer. "Forty spins on spinner A. I predict exactly 20 blue."

"Exactly?" Nova asks. "Or about?"

"About," Rocket says. "Half of 40 is 20. Ten red, ten green."

Raven keeps the tally. The pointer whirs. Tick marks pile up. "What do you notice?" she asks at the end.

Rocket reads the sheet. "Twenty-two blue. Nine red. Nine green. Close, but not exact."

"Probability predicts the long run," Nova says. "Not any single spin, and not an exact count."

"So 22 out of 40 is the relative frequency," Raven says. "And 1/2 is the probability. They should be close."

Rocket grins. "Close enough. Spinner A is fair."

Predicting a count

Probability tells you the share of tries you expect in the long run. Multiply the probability by the number of tries.

Blue has probability 1/2. In 40 spins, you expect about 1/2 × 40 = 20 blue.

Red has probability 1/4. Expect about 1/4 × 40 = 10 red, and about 10 green.

These are predictions, not promises. Real spins wobble around the predicted count. A few more here, a few fewer there.

Checking against real spins

Raven tallied 40 spins of spinner A. Here is the prediction beside what actually happened.

The relative frequency of an event is the times it happened divided by the total tries. It is a fraction of the spins.

Blue came up 22 times in 40 spins: a relative frequency of 22/40 = 11/20. The probability was 1/2. Close.

When the observed counts sit near the predicted counts, the model fits. When they sit far off, something about the spinner or the spinning needs a look.

ColorProbabilityPredicted in 40 spinsObservedRelative frequency
red1/41099/40
blue1/2202211/20
green1/41099/40
Spinner A: four equal sectors, 1 red, 2 blue and 1 green, with a pointer.

Build a spinner

  1. Draw lines across a paper plate to split it into equal sectors. Fold it in half, then in half again, for four.
  2. Color each sector. Equal sectors make equally likely outcomes.
  3. Lay a paper clip on the plate and hold a pencil point through its end at the center.
  4. Flick the free end of the clip. Where it stops is the outcome.
  5. Tally each outcome. Compare the tally with your prediction when you finish.
PREDICT THE COUNT
  • Read the question.
  • Tap your answer.
Spinner AThe probability is 0.5. In 40 tries, about how many times should blue happen?
Spinner AThe probability is 0.25. In 40 tries, about how many times should red happen?
The probability of green is 1/4. In 100 spins, about how many greens should there be?
The probability of blue is 1/2. In 200 spins, about how many blues should there be?
COMPARE WITH THE SPINS
  • Read the question.
  • Tap your answer.
Blue was predicted 20 times and came up 22 times. How many more than predicted?
Red came up 9 times in 40 spins. What is the relative frequency of red, in lowest terms?
Add the observed counts for red, blue and green. What do they total?
Observed 22 blue, predicted 20. What does this say about the spinner?
TEST A SPINNER
  • ?Decide whether the model fits or needs a second look.
  • ?Spin and tally the real outcomes.
  • ?Multiply each probability by the number of spins to predict a count.
  • ?Compare the observed counts with the predicted counts.
  • ?Find the probability of each color from the sectors.
WHY THIS EXERCISEThis is the test you will run on your own Data Fair spinner.
Predicted greens in 40 spins
Observed greens in Raven's 40 spins
The probabilities of red, blue and green add to
Try it
Say what you would expect in 400 spins of spinner A. Then say why 199 blue would not surprise you.

Predict, spin, compare. Tomorrow you build your own spinner and run the test yourself.

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