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Statistics 9-12 / Week 11 / Wednesday
3/6
Week 11 · Counting and the Multiplication Rule

Wednesday

Data Lab: twenty double draws
// Volunteer shifts drawn two at a time
⏱ about 20 min

Wednesday: Data Lab, Twenty Double Draws

"Draw two, count the reds, put them back, shake," Comet says. "Twenty times. Wren keeps the tally."

She draws. Red, blue. "One red." Again. Red, red. "Two." Again. Gold, blue. "Zero."

After twenty rounds Wren reads the tally. "Zero reds 3 times, one red 9 times, two reds 8 times."

"What do you notice?" he asks. "One red is the common case."

Nova projects three bars. "Would you like a hint? Work out what the bag predicts, then compare."

"Two reds is 5/8 × 4/7 = 5/14," Comet says. "Out of twenty, that predicts about 7.1."

"We got 8," Wren says. "Close. Twenty rounds is not forever, and the bag does not owe us anything."

What you need

  • A cloth bag or a cup and 8 tokens or paper scraps: 5 marked R, 2 marked B, 1 marked G.
  • Your Run Log and a pencil.
  • A partner to keep the tally, and a grown-up aware of where you are.
Safety first
The lab stays at a table. Tokens and paper scraps stay on the table and away from small children.
Cut paper scraps with scissors on paper or cardboard, never toward a hand.
Any jogging this week happens on the park loop or a safe path, with a grown-up aware and water at hand. No one runs on a road.
Nothing heavy is lifted alone.

What the bag predicts

Two draws without replacement make 56 equally likely ordered pairs. Count the pairs for each number of reds.

Zero reds: 3 × 2 = 6 pairs, so P = 3/28. Two reds: 5 × 4 = 20 pairs, so P = 5/14.

One red: red then other, or other then red. 5 × 3 twice is 30 pairs, so P = 15/28.

Check: 3/28 + 15/28 + 5/14 = 1. Every pair is counted once.

Run the lab

  1. Put 8 tokens in the bag: 5 R, 2 B, 1 G. Shake it.
  2. Draw two tokens without putting the first back. Count the reds: 0, 1 or 2. Tally it.
  3. Put both tokens back and shake. Repeat until you have 20 rounds.
  4. Turn each tally into a fraction of 20. Write the bag's probabilities beside them: 3/28, 15/28, 5/14.
  5. Multiply each probability by 20 to get the predicted counts. Compare with your tally, bar by bar.

The crew's lab log

Here is the crew's own tally from twenty rounds. Yours will differ, and that is the point of a lab.

Reds in two drawsTally (of 20)Tally as a fractionProbability from the bagPredicted count
033/203/282.1
199/2015/2810.7
282/55/147.1

The predicted counts are the bag's probabilities times 20. The crew's tally landed close to each, not on it.

Two reds has a second route: C(5, 2) = 10 red pairs out of C(8, 2) = 28 pairs, 5/14. Same answer as the rule.

READ THE LAB LOG
  • Read the question.
  • Tap your answer.
The bag holds 5 red, 2 blue and 1 gold. Two are drawn without replacement. What is the probability both are red?
A bag holds 8 tokens, 5 of them red tokens. 2 are drawn at once. What is the probability all 2 are red tokens?
In 20 rounds, about how many rounds with two reds does the probability 5/14 predict? (Round to 1 place.)
The crew's tally did not match the prediction exactly. What does that show?
In the crew's run, how many rounds gave exactly one red? Type the number.
WHY THIS EXERCISEOne red can happen two ways, red then other or other then red. That is why its probability is the largest.
What the lab showsTrue or false?
3 + 9 + 8 = 20?
3/28 + 15/28 + 5/14 = 1?
A tally of twenty rounds must match the bag's probabilities exactly.?
Two reds can be counted by ordered pairs or by combinations, and both give the same probability.?
WHY THIS EXERCISEThe lab shows the rule and the tally side by side, and why they differ a little.
Draw your tally as three bars beside three bars of the bag's predicted counts.

Careful lab work. Tomorrow you prove the multiplication rule by counting pairs and use combinations for "at least one".

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