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Statistics 9-12 / Week 11 / Track Day
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Week 11 · Counting and the Multiplication Rule

Track Day

Track Day: the family shift draw
// Volunteer shifts drawn two at a time
⏱ about 20 min

Track Day: The Family Shift Draw

Saturday, five scraps go into a cup: 2 marked R and 3 marked B.

"Two chores, two draws," Comet says. "The first scrap stays out. Chance both are R?"

"R first is 2/5," Wren says. "Then only 1 R is left among 4. Multiply."

"2/5 × 1/4 = 1/10," someone says. "One in ten."

"Check it with groups," Comet says. "C(5, 2) = 10 pairs, and only one pair is both R. 1/10."

"What do you notice?" Wren asks. "Two routes, one answer, every time."

Nova hovers over the cup. "Would you like a hint? Draw ten times and tally before you decide if it feels right."

  1. Without replacement, P(A and B) = P(A) × P(B given A). The second fraction comes from the smaller bag.
  2. The rule runs both ways: P(B) × P(A given B) gives the same answer. With replacement the draws are independent.
  3. Permutations count orders and combinations count groups: C(n, k) = P(n, k) ÷ k!.
  4. Equally likely orders or groups turn into probabilities: favorable over all.
  5. P(at least one) = 1 - P(none). Count none first.
◇ FAMILY SHIFT DRAW
Put 5 scraps in a cup: 2 marked R and 3 marked B.
Each person finds P(both R) by the rule, then by counting the pairs. The answers must match.
Find P(at least one R) together by counting the chance of no R first.
Draw two scraps without replacement ten times. Tally both-R, one-R and no-R.
Compare the tally with the three probabilities. Say in everyday words why they differ a little.
For a grown-up
This is a table activity with paper scraps. Scraps stay on the table and away from small children.
If the two routes disagree, recount the pairs together. The counting is the lesson.
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.
A cup holds 2 R scraps and 3 B scraps. Two are drawn without replacement. What is the probability both are R?
How many different pairs of scraps can be drawn from the 5?
A bag holds 5 tokens, 2 of them R scraps. 2 are drawn at once. What is the probability of at least one R scrap?
After one R scrap is drawn and kept out, how many R scraps are left in the cup? Type the number.
WHY THIS EXERCISEThat smaller count is the top of the second fraction. It is what "without replacement" changes.
Draw the cup before and after the first draw. Write the two fractions and their product.
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