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Precalculus 9-12 / Week 12 / Thursday
4/6
Week 12 · Counting, Expected Value and Regatta Day

Thursday

Weighing a decision
// The heats drawn by lot
⏱ about 20 min

Thursday: Weighing a Decision

"Points on Regatta Day," Comet says, pinning a card. "5 for first, 3 for second, 1 for third."

"The practice log says Kestrel finishes first half the time, second a third, third a sixth," Wren says.

"So how many points should Kestrel expect in a heat?"

Comet multiplies. "5 × 1/2 + 3 × 1/3 + 1 × 1/6. That is 11/3."

"About three and two thirds," Wren says. "Not a score anyone gets. The mean over many heats."

Nova projects two cards, Steady and Sprint. "Would you like a hint? Weigh each strategy the same way."

"Steady gets 3. Sprint gets 2.6," Comet says after a minute. "Steady, by a little."

"What do you notice? Sprint wins more often than it loses, and still scores less on average."

Expected value weighs a decision

A decision with chance in it has several outcomes, each with a value and a probability.

Its expected value is each value times its probability, added. It is the long-run mean score of that choice.

To compare two strategies, find each expected value and pick the larger. The crew calls this weighing the decision.

Kestrel's heat: 5 × 1/2 + 3 × 1/3 + 1 × 1/6 = 11/3 points per heat, in the long run.

A solved problem: Steady or Sprint

  1. Steady: first, second and third each with probability 1/3, from the crew's practice log.
  2. Expected points: 5 × 1/3 + 3 × 1/3 + 1 × 1/3 = 3.
  3. Sprint: first with probability 2/5, third with probability 3/5, never second.
  4. Expected points: 5 × 2/5 + 1 × 3/5 = 2.6.
  5. Compare: 3 is more than 2.6. Over many heats, Steady scores more.
  6. Sprint is exciting and often first. Expected value says the slow third places drag it down.
Steady's distribution: 5, 3, 1 points each with probability 1/3.

Why P(A and B) = P(A) × P(B given A)

Monday's rule, proved by counting the ordered pairs of draws. Read it first, then put it in order.

  1. The bag has 6 tokens, so two draws make 6 × 5 = 30 ordered pairs, all equally likely.
  2. Pairs with a red first: 3 × 5 = 15. So P(A) = 15/30.
  3. Among those 15 pairs, the ones with a blue second: 3 × 2 = 6. So P(B given A) = 6/15.
  4. Pairs with red first and blue second are those same 6, out of all 30. So P(A and B) = 6/30.
  5. Multiply: 15/30 × 6/15 = 6/30. The 15 cancels.
  6. In general: (A pairs ÷ all) × (A and B pairs ÷ A pairs) = A and B pairs ÷ all. That is the rule.
THE PROOF, IN ORDER
  • ?Two draws make 30 equally likely ordered pairs
  • ?6 of those have a blue second, so P(B given A) = 6/15
  • ?15/30 × 6/15 = 6/30, the shared 15 cancels
  • ?Those 6 pairs are the A and B pairs, so P(A and B) = 6/30
  • ?15 pairs have a red first, so P(A) = 15/30
  • ?So P(A and B) = P(A) × P(B given A)
WHY THIS EXERCISEThe rule is one cancelling fraction. Counting the pairs is what makes it true.
EXPECTED POINTS AND STRATEGIES
  • Read the question.
  • Tap your answer.
A bar chart of a probability distribution: values 5, 3, 1 with probabilities 1/2, 1/3, 1/6Kestrel scores 5 points with probability 1/2, 3 with probability 1/3, 1 with probability 1/6. What is the expected value, in points?
Option Steady scores 5 with probability 1/3, 3 with probability 1/3, 1 with probability 1/3. Option Sprint scores 5 with probability 2/5, 1 with probability 3/5. Which option has the higher expected value?
Light oars: 5 points with probability 1/2, 1 with probability 1/2. Heavy oars: 5, 3, 1 with probabilities 1/4, 1/4, 1/2. Which has the higher expected value?
WHICH SOLUTION MAKES SENSE?
  • Read the question.
  • Tap your answer.
Sprint finishes first 2/5 of the time and Steady only 1/3. Why does Steady have the higher expected value?
Kestrel's expected value is 11/3 points. What does that number mean?
Each value times its probability, added up, is called the this value. Type one word.
What is Steady's expected value, in points? Type the number.
P(B given A) is called a this probability. Type one word.
StatementTrue or false?
Expected value is the long-run mean of a random variable.?
Steady's expected value of 3 beats Sprint's 2.6.?
The strategy that wins most often always has the higher expected value.?
5 × 2 + 1 × 3 = 13?
An expected value of 11/3 points means a boat can score 11/3 in one heat.?
WHY THIS EXERCISEWeighing by probability is what expected value adds to a plain average.
Try it
Make up a third strategy with its own probabilities for first, second and third. The probabilities must add to 1.
Find its expected points and rank all three strategies.

Sharp thinking. Tomorrow the crew makes the lane draw fair and plans the Regatta Day table.

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