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Statistics 9-12 / Week 10 / Track Day
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Week 10 · Probability Rules

Track Day

Track Day: the family lane draw
// The lane draw, with and, or and not
⏱ about 20 min

Track Day: The Family Lane Draw

Saturday morning, six scraps of paper numbered 1 to 6 go into a cup on the kitchen table.

"Two events," Comet says. "Even. And small, meaning 3 or less. Every scrap lands in one box of the grid."

"Pick one with your eyes shut," Wren says. "What is the chance it is even?"

"1/2," someone answers. "And even and small?"

"Only the 2," Comet says. "1/6. And even or small is everything but the 5."

"What do you notice?" Wren asks. "P(even) times P(small) is 1/4, not 1/6. Not independent in this cup."

Nova hovers over the cup. "Would you like a hint? Try the addition rule before you count."

  1. The sample space is every outcome. An event is a set of outcomes: "and" keeps the overlap, "or" joins, "not" takes the rest.
  2. P(A or B) = P(A) + P(B) - P(A and B): the overlap is counted once. Disjoint events have nothing to subtract.
  3. P(A given B) = outcomes in both ÷ outcomes in B = P(A and B) ÷ P(B). The group total goes on the bottom.
  4. Independent means P(A given B) = P(A), the same as P(A and B) = P(A) × P(B).
  5. Everyday sentences hide a "given". Find the group named after "among" or "of".
◇ FAMILY LANE DRAW
Put six scraps numbered 1 to 6 in a cup. Agree on two events, such as even and 3-or-less.
Each person lists the outcomes in A, in B, in A and B, in A or B and in not A.
Then each person writes the five probabilities as fractions in lowest terms.
Check P(A or B) two ways: by the list and by the addition rule.
Run the product test together. Are your two events independent in this cup?
Draw ten times with replacement and tally. Compare your tally with the probabilities you wrote.
For a grown-up
This is a table activity with paper scraps. Scraps stay on the table and away from small children.
A disagreement about a count is a good moment: list the outcomes together and recount.
The numbers belong to your cup only, like the crew's bag. No fact about anything else is being claimed.
TRACK DAY CHECK
  • Read the question.
  • Tap your answer.
Tokens 1 to 6 are equally likely. Even = {2, 4, 6} and small = {1, 2, 3}. What is P(even and small)?
Tokens 1 to 6 are equally likely. Even = {2, 4, 6} and small = {1, 2, 3}. What is P(even or small)?
P(even) = 1/2, P(small) = 1/2 and P(even and small) = 1/6. Are the events independent?
In the cup of six, how many scraps are both even and small (3 or less)? Type the number.
WHY THIS EXERCISEThe overlap is the top of every "and" and every "given" fraction. Count it first.
Draw your six scraps in a row. Loop the even ones and the small ones. Write the five probabilities under them.
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