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Classroom activity Grade 12 Math

Verrindale Leak Survey Round: Ranking Segments by Expected Value

A 45-minute grade 12 team round where students compute expected water-loss recovery per survey dollar and choose a leak survey plan under a fixed budget.

Objectives

  1. Students will compute the expected value of a leak survey from a probability and a stated loss amount.
  2. Students will rank candidate segments by expected recovery per survey dollar.
  3. Students will select a viable survey plan under a fixed budget and defend the ranking they used.

Vocabulary

Non-revenue water
Water that a utility treats and pumps but never gets paid for. Most of it escapes through leaks in old pipes, and some is lost to broken meters or theft.

Also: expected value, leak survey, budget constraint, ranking

Teacher overview

A utility that suspects leaks in many places and can afford to survey only a few must decide which places to survey first. This round puts teams of three in charge of the fictional Verrindale Water District's leak survey budget for one week, with eight candidate main segments on cards. Each card carries a survey cost, an estimated probability that a leak is present, and the annual value of the water that segment would lose if a leak is there. Teams compute expected recovery, divide it by survey cost, rank the eight, and then choose the set that fits the budget. The reasoning students most often need corrected is the assumption that the highest leak probability deserves the first crew: probability alone ignores both the cost of looking and the size of what is being lost.

Activity Directions

  1. Setup (5 min): Form teams of three and give each team the eight segment cards from the fictional Verrindale system, a ranking sheet, and a calculator. Write the two working definitions on the board: expected recovery equals the probability of a leak times the annual value lost if a leak is present, and the ranking quantity is expected recovery divided by survey cost.
  2. Assign roles (2 min): In each team, one student computes expected recovery, one computes expected recovery per survey dollar, and one records and checks. Roles rotate after the fourth card so every student does each job.
  3. Round 1, compute the cards (12 min): Teams work through all eight fictional Verrindale cards and complete questions 1 through 7 on the handout. A card is finished only when both figures are written on the ranking sheet. Stop the round at 12 minutes and check that every team has all eight cards scored.
  4. Round 2, spend the budget (10 min): Announce the survey budget of $6,000 for the week. Teams answer questions 8 through 11 by choosing the set of segments that maximizes total expected recovery without exceeding the budget. Each team writes its chosen set on the board when it finishes.
  5. Scoring (4 min): Award two points for each correct expected recovery figure a team can defend, and five points for identifying the highest-value set that fits the budget. Read out the totals and confirm which sets on the board reach $25,000 of expected recovery.
  6. Debrief (12 min): Ask three questions in turn, taking responses from at least two teams each. First, why does the segment with the lowest leak probability still earn a place in the plan for the fictional district. Second, why is a plan that spends every dollar of the budget not automatically the best plan. Third, what would change in the ranking if the district could survey a segment for half the listed cost using a new acoustic device, and which segments would move most.
Verrindale Leak Survey Round: Ranking Segments by Expected Value — Grade 12 Math
Name: ____________________________ Date: ______________

Student Handout

The fictional Verrindale Water District can survey some of its eight aging main segments this week. Each segment lists the cost of surveying it, the estimated probability that a leak is present, and the annual value of the water that would be lost if a leak is there and goes unfound. Expected recovery means the probability times that annual value.

Segment Survey cost Probability of a leak Annual value lost if a leak is present
S1 $1,800 0.40 $14,000
S2 $1,200 0.25 $16,000
S3 $2,400 0.55 $12,000
S4 $900 0.15 $20,000
S5 $3,000 0.60 $25,000
S6 $1,500 0.20 $9,000
S7 $2,100 0.35 $18,000
S8 $600 0.10 $30,000
  1. Compute the expected recovery for segment S1.
  2. Compute the expected recovery for segment S5.
  3. Compute the expected recovery for segment S8.
  4. Compute the expected recovery per survey dollar for segment S5. Round to two decimal places.
  5. Compute the expected recovery per survey dollar for segment S8. Round to two decimal places.
  6. Compute the expected recovery per survey dollar for segment S6. Round to two decimal places.
  7. Which one of the eight segments has the lowest expected recovery per survey dollar?
  8. The survey budget is $6,000. Which set of segments gives the largest total expected recovery without exceeding it?
  9. What is the total survey cost of the set you chose in question 8?
  10. What is the total expected recovery of that set?
  11. How many dollars of the $6,000 budget does that set leave unspent?
  12. Segment S8 has the lowest leak probability of all eight. Explain in two or three sentences why it still belongs in the chosen set, while segment S3, with the second-highest probability, does not.
  13. Explain in two or three sentences why the district should expect to find no leak on some of the segments it surveys, even though every choice was made to maximize expected recovery.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key

  1. Answer: $5,600.
    Segment S1 has a 0.40 probability against a $14,000 annual loss, so the expected recovery is 0.40 times 14,000, or $5,600.

  2. Answer: $15,000.
    Segment S5 has a 0.60 probability against a $25,000 annual loss, so 0.60 times 25,000 gives $15,000.

  3. Answer: $3,000.
    Segment S8 has a 0.10 probability against a $30,000 annual loss, so 0.10 times 30,000 gives $3,000.

  4. Answer: 5.00 dollars per dollar.
    Dividing S5's expected recovery of $15,000 by its $3,000 survey cost gives 5.00 dollars of expected recovery for every survey dollar.

  5. Answer: 5.00 dollars per dollar.
    Dividing S8's expected recovery of $3,000 by its $600 survey cost also gives 5.00, tying S5 at the top of the ranking despite the much smaller expected recovery.

  6. Answer: 1.20 dollars per dollar.
    S6's expected recovery is 0.20 times $9,000, or $1,800, and dividing by the $1,500 survey cost gives 1.20.

  7. Answer: Segment S6.
    Its ratio of 1.20 is the lowest of the eight. The next lowest is S3 at 2.75, then S7 at 3.00, S1 at 3.11, S2 and S4 at 3.33, and S5 and S8 tied at 5.00.

  8. Answer: Segments S5, S8, S2, and S4.
    Taking the segments in ranking order fills the budget with S5 at $3,000, S8 at $600, S2 at $1,200, and S4 at $900. No other combination within $6,000 reaches a higher total expected recovery.

  9. Answer: $5,700.
    Adding $3,000, $600, $1,200, and $900 gives a total survey cost of $5,700.

  10. Answer: $25,000.
    Adding the expected recoveries of $15,000, $3,000, $4,000, and $3,000 gives $25,000.

  11. Answer: $300.
    The $6,000 budget less the $5,700 spent leaves $300, which is not enough to add any remaining segment, since the cheapest unchosen card costs $1,500.

  12. Answer: S8 is cheap to survey and protects a large loss, while S3 is expensive and protects a small one.
    S8 costs only $600 and guards $30,000 a year, so each survey dollar buys 5.00 dollars of expected recovery. S3 costs $2,400 and guards $12,000, so even with the second-highest probability on the sheet its ratio is only 2.75. Probability is one factor of three, and by itself it does not decide the ranking.

  13. Answer: Expected value describes an average over many surveys, not the outcome of any single one.
    Every chosen segment has a probability below 1, so on any given week some crews will open a street and find nothing. The plan is defensible because repeating this ranking across many weeks recovers more water per survey dollar than ranking by leak probability alone, not because each individual survey succeeds.

Differentiation

Support

Give these students a ranking sheet for the fictional Verrindale cards with the expected recovery column already filled in, so their work is the division and the ordering rather than both steps. Provide one worked card from outside the set, showing the probability, the loss, the product, and the ratio in a labeled column, and let them work with four cards instead of eight in round 1.

Extension

Ask students to redo the budget round for the fictional district under a rule that no more than two segments may be surveyed in any one week, and report how much expected recovery that restriction costs. Then have them find the survey cost at which segment S3 would move ahead of segment S2 in the ranking, and explain what a cheaper detection method would do to the whole plan.

ELL supports

Pre-teach "expected value," "probability," "budget constraint," and "unspent" with a labeled card showing the three numbers and arrows to the two figures students must compute. Provide sentence frames for items 12 and 13, such as "Segment S8 is chosen because its cost is ___ and its possible loss is ___," and allow teams to present their chosen set as a marked-up table before writing the explanations.

Extensions

Teams can add a ninth card to the fictional Verrindale set with numbers of their own choosing, designed so that it displaces exactly one segment from the winning plan, and then justify their design. A second follow-up asks students to describe how a district would estimate the probability figures on these cards in the first place, using pipe age, past break records, and pressure data, and what would make such an estimate unreliable.