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Statistics 9-12 / Week 07 / Monday
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Week 07 · Does the Model Fit?

Monday

A model and a run of tails
// The pacing trial and a run of tails
⏱ about 20 min

Monday: A Model and a Run of Tails

The pacing trial starts at the park loop. Before each practice lap, a coin decides who leads.

"Heads, Comet leads. Tails, I lead," Wren says, flipping over the tally sheet.

Tails. Tails. Tails. Tails. Tails. Comet stares at the row. "5 tails in a row. Is this coin fair?"

"The model says heads half the time," Wren says. "What do you notice? We have not tested it yet."

Nova hovers up a bar graph of possible head counts. "Would you like a hint? Count the orders for 5 flips."

"32 orders, all equally likely," Comet says. "Only one is all tails. That is 1/32."

"Rare, but not impossible," Wren says. "A fair coin does this about once in thirty-two tries."

"Then we keep flipping and keep counting," Comet says. "The whole run will tell us more than five flips."

Wren flips a coin over a tally sheet, Comet eyes a row of five tails, Nova projects a bar graph.

The model and the question

A probability model is a claim about chance. The crew's model: a fair coin lands heads with probability 1/2, and every flip is independent.

This week asks one question of every result: could this model easily produce it? If yes, the result is consistent with the model.

If the result would almost never happen under the model, the crew questions the model, not the result.

Every flip in this course is the crew's own reading from Nova's Run Log, written for this lesson.

A solved problem to study

Here is how Wren finds the probability of exactly 2 heads in 5 fair flips.

  1. Each flip has 2 outcomes, so 5 flips have 2 × 2 × 2 × 2 × 2 = 32 orders. The fair coin makes them equally likely.
  2. Count the orders with exactly 2 heads: choose which 2 of the 5 flips are heads. C(5, 2) = 10.
  3. Divide: 10 ÷ 32 = 5/16.
  4. Check with the formula: C(5, 2) × (1/2)^2 × (1/2)^3 = 10 × 1/32 = 5/16.
  5. All 5 tails is one order of 32: 1/32. So is all 5 heads.

The formula says: count the orders with k successes. Then multiply by p for each success and 1 - p for each failure.

Heads in 5 flipsOrdersProbability
011/32
155/32
21010/32
31010/32
455/32
511/32
A bar graph of the crew's model for 5 fair flips: the tallest bar is at 2 and 3 heads.
COUNT THE ORDERS
  • Read the question.
  • Tap your answer.
A bar graph of a binomial model with 5 tries and success probability 1/2: the tallest bar is at 2 successes.Comet flips a fair coin 5 times. What is the probability of exactly 2 heads?
A bar graph of a binomial model with 5 tries and success probability 1/2: the tallest bar is at 2 successes.Wren flips a fair coin 5 times. What is the probability of exactly 1 head?
The model says heads has probability 1/2. What is the probability of 5 tails in a row?
How many equally likely orders do 5 coin flips have? Type the number.
WHY THIS EXERCISEEvery probability this week is a count of orders over this total. The total comes first.
StatementTrue or false?
10 ÷ 32 = 5/16?
Five tails in a row has probability 1/32 under the fair-coin model.?
A rare result proves the model is wrong.?
Each flip of a fair coin is independent of the flips before it.?
WHY THIS EXERCISEThe crew judges a model by how rare the result is. A rare result alone is only a reason to look harder.
Try it
On an index card, list all 8 orders of 3 coin flips (HHH, HHT, and so on). Circle the orders with exactly 2 heads.
Count them and write the probability as a fraction. Compare with C(3, 2) ÷ 8.
Draw a tree of the 8 orders of 3 coin flips. Circle the branches with exactly 2 heads.

A strong start. Tomorrow you build the whole distribution two ways and find the count the model expects.