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From Outage to Refund: Calculating Expected Compensation Using Probability

UUnknown

2026-03-03

10 min read

Learn to compute expected refunds and fair compensation after outages with a step-by-step probability worksheet tied to the Verizon $20 credit case.

Hook: When an Outage Costs You Time — and Grades

You pulled an all-nighter to finish a group project, you rely on your phone for school alerts, and then — outage. Students, teachers, and lifelong learners: outages don't just interrupt streaming or calls; they create real academic stress. How much should a company like Verizon refund customers after a major service disruption? More importantly for learners: how do you compute a fair expected compensation using probability?

Why this worksheet matters in 2026

In late 2025 and into 2026, telecom reliability and automatic compensation policies moved from opinion pieces into public policy debate. Major providers have started offering automated credits (for example, a widely reported $20 credit after a major Verizon outage), while regulators and consumer groups pushed for clearer, data-driven refund rules. For anyone studying probability or preparing for exams, this is a great real-world application: you can use expected value, distributions, and variance to design and evaluate compensation policies that balance customer fairness and company risk.

Learning objectives

Model an outage distribution with a discrete probability mass function (PMF).
Compute the expected refund under different refund policies.
Compare fairness metrics (per-customer vs per-affected-customer).
Estimate company expected costs and variability (variance and standard deviation).
Practice hypothesis-driven policy design: flat credit vs proportional refund vs capped per-hour refund.

Quick refresher: Expected value and why it matters

The expected value E[X] of a random variable X (here, refund paid to a customer) is the long-run average if a policy is repeated many times. For discrete outcomes, E[X] = sum p_i * x_i. Companies use expected value to budget for outages; students use it to solve exam-style questions and real policy problems.

Step-by-step worksheet: From outage distribution to expected refund

We’ll walk through a complete worked example. Keep a calculator handy.

Step 1 — Define a plausible outage distribution

Suppose historical monitoring in a region yields this estimated discrete distribution for outage length L (hours):

P(L = 0) = 0.70 (no outage)
P(0 < L ≤ 1) = 0.10
P(1 < L ≤ 3) = 0.12
P(3 < L ≤ 8) = 0.06
P(L > 8) = 0.02

This distribution is realistic for tutorial purposes: outages are rare (70% no outage) but can be severe occasionally.

Step 2 — Policy A: Flat credit (example: the $20 credit)

Many providers have offered a flat credit after a major outage (for instance, Verizon's $20 credit in response to a high-profile disruption). Under a common interpretation, the company credits affected customers — those with L > 0 — a flat $20.

Let X be the refund to a randomly chosen customer. Then X = 20 with probability P(L > 0) = 1 - 0.70 = 0.30, and X = 0 otherwise. The expected refund per customer is:

E[X] = 0.30 * 20 + 0.70 * 0 = $6.00

So on average, the company should expect to pay $6 per customer in this region. For a million customers, that's an expected payout of $6,000,000.

Step 3 — Policy B: Proportional per-hour refund with a cap

An alternative is $5 per hour of outage, capped at $50. To compute the expected refund, assign representative average durations (midpoints) for each positive-length category:

(0, 1] → 0.5 hours average → refund $2.50
(1, 3] → 2 hours average → refund $10.00
(3, 8] → 5.5 hours average → refund $27.50
(>8) → assume 12 hours average → refund $60, but capped at $50 → refund $50

Compute E[X] = sum p_i * refund_i across these positive categories:

0.10 * 2.5 = 0.25
0.12 * 10 = 1.20
0.06 * 27.5 = 1.65
0.02 * 50 = 1.00

Sum E[X] = 0.25 + 1.20 + 1.65 + 1.00 = $4.10 per customer.

This policy yields a lower expected per-customer payout than the flat $20 credit ($4.10 vs $6.00), but it concentrates compensation toward longer outages.

Step 4 — Compare fairness: per-customer vs per-affected-customer

A common fairness metric is expected refund conditional on being affected: E[X | L > 0] = E[X] / P(L > 0).

Flat credit: E[X | affected] = 6.00 / 0.30 = $20.00 (by design).
Proportional credit: E[X | affected] = 4.10 / 0.30 ≈ $13.67.

Interpretation: affected customers get more on average under the flat $20 policy, while the proportional policy targets payouts better toward longer outages but pays less on average to affected customers.

Step 5 — Variance and predictability

Companies care about variance because budget swings are risky. Compute Var(X) = E[X^2] - (E[X])^2.

For the flat credit (binary): E[X^2] = 0.30 * 20^2 = 0.30 * 400 = 120. Var = 120 - 6^2 = 120 - 36 = 84. Standard deviation ≈ sqrt(84) ≈ 9.17.

For the proportional policy, compute E[X^2] by summing p * refund^2:

0.10 * (2.5)^2 = 0.10 * 6.25 = 0.625
0.12 * 100 = 12.00
0.06 * 756.25 = 45.375
0.02 * 2500 = 50.00

E[X^2] = 0.625 + 12 + 45.375 + 50 = 108. Var = 108 - (4.1)^2 = 108 - 16.81 = 91.19. SD ≈ 9.55.

Both policies show similar standard deviation, so predictability is comparable. The proportional policy slightly increases variance in this example because it creates a wider range of outcomes.

Practice problems (worksheet)

Try these independently — solutions are provided after the problems so you can check your work.

Problem 1 (Basic):

Given the same outage PMF above, compute the expected company payout for a region with 250,000 customers under:

Flat $20 credit to all affected customers (Policy A).
$10 per hour, no cap, using the midpoints above for averages.

Problem 2 (Intermediate):

You observe 100 outage events with these durations (hours): 0,0,0,0.5,1,2,0,0,4,10,... (assume a sample mean of 2 hours for positive events). Fit an exponential distribution to the positive outage durations (exponential MLE: lambda = 1/mean). Using a proportional policy of $5/hr with no cap, compute the expected refund by integrating the exponential distribution (E[refund] = integral over t>0 of 5t * f(t) dt, or use E[T] times 5).

Problem 3 (Advanced):

Design a refund policy that meets both goals:

Expected revenue loss per customer ≤ $5
Expected refund among affected customers ≥ $15

Use the outage PMF above. Propose a policy (flat + per-hour hybrid, or tiered) and compute E[X] and E[X | affected] to verify constraints.

Solutions (worked)

Solution 1

Policy A: E[X] = $6 per customer. For 250,000 customers, expected payout = 6 * 250,000 = $1,500,000.
$10/hr no cap: Using the same midpoints: refunds would be 5, 20, 55, 120 for the four positive categories (0.5*10 =5, 2*10=20, 5.5*10=55, 12*10=120). E[X] = 0.10*5 + 0.12*20 + 0.06*55 + 0.02*120 = 0.5 + 2.4 + 3.3 + 2.4 = $8.6 per customer. Total payout = 8.6 * 250,000 = $2,150,000.

Solution 2

If positive outage durations follow exponential with mean 2 hours, lambda = 1/2 = 0.5. E[T] = 2. For refund = 5 * T, E[refund] = 5 * E[T] = 10. That is, average refund = $10 for affected customers. To get per-customer expected refund multiply by P(affected). If overall P(affected) is still 0.30, E[X] = 0.30 * 10 = $3.00 per customer.

Solution 3 (design example)

Try hybrid policy: $10 flat when affected + $2 per hour for hours beyond 1, capped at $60 total. Approximate expected refund:

For (0,1]: refund = $10 (no extra hours). Contribution: 0.10*10 = 1.0
For (1,3]: average 2 hrs => extra 1 hr → extra $2 → total 12. Contribution: 0.12*12 = 1.44
For (3,8]: average 5.5 hrs => extra 4.5 hrs → extra $9 → total 19. Contribution: 0.06*19 = 1.14
(>8): average 12 hrs => extra 11 hrs → extra $22 but capping at $60 total means if base 10 + 22 = 32 < 60, use 32; Contribution: 0.02*32 = 0.64

Sum E[X] = 1.0 + 1.44 + 1.14 + 0.64 = $4.22 per customer. E[X | affected] = 4.22 / 0.30 ≈ $14.07. This meets the expected revenue loss constraint (≤$5) but misses the affected-customer target (≥$15). Increase base or per-hour rate until both constraints met — this is a standard optimization exercise using the PMF.

Advanced strategies and 2026 trends

As of 2026, three trends are important for students and policy-minded problem solvers:

Automated detection and instant credits: AI-driven monitoring now allows many providers to detect outages in near real-time and issue automated credits. This reduces administrative friction and supports per-minute proportional refunds in some pilot programs.
Data-driven regulation: Regulators and consumer advocates (especially after late-2025 high-profile outages) demanded clearer compensation formulas. The result: more transparent SLA-style policies, though final national-level rules remain under discussion in several jurisdictions as of 2026.
Risk-sharing approaches: Insurers and large providers are exploring outage insurance and pooled reserves, which changes the expected-value calculus companies use when setting refunds.

For students: these trends make probabilistic modeling even more relevant. Companies now model payouts using Monte Carlo simulations across many regions and time windows; you can replicate that for practice using simple Python or spreadsheet simulations.

Monte Carlo mini-lab (optional)

Want to simulate 100,000 customers' refunds to estimate company budgets and tail risks? Use this simple approach:

Generate a random uniform number u in [0,1].
Map u to a category using cumulative probabilities from the PMF.
Assign a duration (use midpoint or sample from a continuous fit like exponential if you estimated one).
Compute refund according to the policy rule.
Aggregate refunds and compute mean, median, 95th percentile, and variance.

This lab helps you see not only expected payouts but also the probability of extreme budget outcomes — a key element of policy design.

Practical takeaways for students and educators

Model first, then design: Always start from an outage distribution before deciding on flat or proportional refunds.
Use conditional expectations: E[X | affected] reveals the fairness experienced by those who actually suffered outages.
Balance generosity and predictability: Flat credits are simple and often generous; per-hour schemes are targeted but may reduce perceived fairness for many affected users.
Simulate for stress-testing: Monte Carlo helps detect extreme costs that expected value alone hides.
Connect to policy debates: Understanding expected compensation helps inform regulatory discussions and consumer advocacy — a useful bridge between math class and civic literacy in 2026.

"Calculating expected compensation is a practical exercise in fairness, budgeting, and risk management — perfect for real-world probability practice."

Wrap-up and call-to-action

This worksheet showed how to go from an outage PMF to expected refund, variance, and fairness metrics — all using core probability tools. Whether you're preparing for an exam, designing a mock company policy, or evaluating consumer fairness arguments after events like the 2025 Verizon disruption, these methods scale.

Ready to practice? Download the full worksheet pack with editable spreadsheets, Monte Carlo starter code (Python/Google Sheets), and extra graded problems — or try our interactive expected-value calculator on equations.top.

Join our study community: sign up for weekly practice packs, get new worksheets tied to current 2026 telecom trends, and submit your policy designs for peer review. Turn outage headaches into learning wins.

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