Friday
Friday: Everyday Models
"The warm-up spinner," Comet says, holding up a paper circle in four equal parts. "It picks a station."
"Station 1 should come up 1/4 of the time," Wren says. "Nova, how often did it this month?"
"9 of 20 spins," Nova says. "Would you like a hint? Start with the expected count."
"20 × 1/4 is 5," Comet says. "And the spread is about 1.94. 9 is a long way up."
"More than two spreads," Wren says. "What do you notice about the spinner itself?"
Comet turns it over. A fold in the paper makes one part sit lower. "The model is fine. The spinner is not."
"So the surprise sent us looking, and we found the cause," Wren says. "That is what a model test is for."
"New spinner tomorrow," Comet says. "And this one goes in the Run Log as a lesson."
Models in everyday life
Everyday claims are models too. "This path has a puddle one day in four." "The gate sticks half the time." "The lane draw is fair."
Each claim gives a p. Count the tries n and the successes you saw. Compare with n × p and the spread.
An ordinary result is consistent with the claim. A surprising result sends you looking for a reason, as the fold in the spinner did.
Rare things do happen. Over many checks, about 1 in 20 ordinary results will fall outside the two-spread band by chance alone.
| Everyday claim | p | n | Expected | Spread | Surprising count? |
|---|---|---|---|---|---|
| spinner lands on station 1 | 1/4 | 20 | 5 | 1.94 | 9: yes |
| fair coin lands heads | 1/2 | 10 | 5 | 1.58 | 3: no |
| pacing plan holds the lap | 4/5 | 10 | 8 | 1.26 | 5: yes |
- Read the question.
- Tap your answer.
The spinner model says station 1 has probability 1/4. In 20 spins it came up 9 times. Is that consistent with the model?
A fair four-part spinner is spun 4 times. What is the probability station 1 never comes up?Review: the week in one list
- A probability model gives every outcome a probability. Independent tries have no memory.
- P(exactly k) = C(n, k) × p^k × (1 - p)^(n - k): count the orders, then multiply the probabilities.
- The distribution lists every count with its probability. They add to 1. The expected value is n × p.
- The spread is √(n p (1 - p)). Within two spreads of the expected count is ordinary; beyond is surprising.
- Surprising means doubt the model and look for a reason. It does not prove a new rate.
- Read the question.
- Tap your answer.
A fair coin is flipped 3 times. What is the probability of exactly 2 heads?
A fair coin is flipped 5 times. What is the probability all 5 are heads?| Statement | True or false? |
|---|---|
| 20 × 1/4 = 5 | ? |
| 9 station-1 spins out of 20 is surprising under the fair spinner model. | ? |
| A surprising result always means the model is wrong. | ? |
| About 1 ordinary result in 20 falls outside the two-spread band by chance. | ? |
Compare your count with 5 and the spread 1.94. Is your spinner behaving?
A clear-headed week. Tomorrow is Track Day: your family flips, counts and judges the coin with you.