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Statistics 9-12 / Week 12 / Friday
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Week 12 · Random Variables, Expected Value and Race Day

Friday

A fair draw, and the week in review
// The awards table, built from expected value
⏱ about 20 min

Friday: A Fair Draw, and the Week in Review

"8 runners in the first heat, 8 tokens, one each," Wren says, filling the bag. "Every runner has 1/8 for lane 1."

"What if someone slipped in a second token?" Comet asks. "9 tokens, two for one runner."

"Then that runner has 2/9 and everyone else 1/9," Wren says. "Not fair. The draw favors one person."

Nova projects eight equal bars, then one tall bar. "Would you like a hint? Fair means flat."

"Flat bars, equal chances," Comet says. "And nobody can tell which token they are grabbing."

"Lots are the oldest fair draw there is," Wren says. "A coin for two, a bag of tokens for eight."

"Tomorrow we run the whole awards table," Comet says. "The draw, the pennant chart, the water cups."

"What do you notice? Everything this course ends up on one table beside the loop."

What makes a draw fair

A draw is fair when every outcome has the same probability. 8 tokens, one per runner, give each runner 1/8.

Fair also means no one can steer it: tokens the same size and shape, shaken, drawn without looking.

A coin flip is fair for two choices. A spinner with equal sectors is fair for as many choices as sectors.

Extra tokens break fairness. With two tokens for one runner in 9, that runner has 2/9 and each other runner 1/9.

A bar chart of a fair lane draw: 8 runners, 8 equal bars of probability 1/8.

Review: the week in five lines

  • A random variable gives each outcome a number. Its distribution lists every value with its probability.
  • Theoretical probabilities come from a sample space of equally likely outcomes. Empirical ones come from a tally.
  • Expected value = each value × its probability, added. It is the mean of the distribution.
  • Weigh a decision by comparing expected values. Read a check with a two-way table, not just its rate.
  • A fair draw gives every outcome the same probability and lets no one steer it.
FROM THE RUN LOG
  • Read the question.
  • Tap your answer.
A lane draw gives each of the 8 runners a probability of 1/8. Is the draw fair?
With a second token for one runner, the draw gives that runner 2/9 and each other runner 1/9. Is the draw fair?
A bar chart of a probability distribution: values 2, 1, 0 with probabilities 1/2, 1/4, 1/4A heat awards 2 pennants with probability 1/2, 1 with probability 1/4, 0 with probability 1/4. What is the expected number of pennants?
The pennant bag's expected value is 1.7 per finisher. For 20 finishers, how many pennants should the crew expect?
REASON IT OUT
  • Read the question.
  • Tap your answer.
Which lane draw is fair for 8 runners?
A random variable's probabilities must do what?
A check is right 9 times in 10 on every bike. Does a flag mean the tire is worn 9 times in 10?
With 8 identical tokens, one per runner, what is each runner's probability of lane 1? Type the fraction.
WHY THIS EXERCISEEqual probabilities are the definition of fair. The fraction is the proof.
StatementTrue or false?
A fair draw gives every outcome the same probability.?
Adding a second token for one runner keeps the draw fair.?
8 × 1/8 = 1?
A coin flip is a fair way to decide between two runners.?
A plan is judged by its most likely outcome alone, never by expected value.?
WHY THIS EXERCISEThe review mixes the week: distributions, expected value, decisions and fairness.
Try it
Design a fair draw for your household: one token per person, all identical. Draw for who sets the table.
Then design an unfair one on purpose and say exactly whose probability changed.
Draw the fair draw as 8 equal bars, then the unfair draw with one tall bar. Label each bar's probability.

Fine week's work. Tomorrow is Race Day. The table is yours to run.

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