# Expected value and scenarios > Weighting outcomes by their probabilities, deal pipelines and earn-outs, expected loss against an insurance premium, and risk exposure in matching units. - Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md) - Module: Probability and distributions - Lesson 27/36, ~26 min - Source: https://productovi.com/courses/math-basics/expected-value ## Three ways the summer can go **Alex Tutor:** Back on Klasna's product team: the school is thinking about a summer intensive course. Hanna sketched three scenarios for its profit: a good summer, +300,000 UAH; a normal one, +120,000; a bad one, −80,000. **Sofia:** So which number goes into the plan? **Alex Tutor:** None of them alone. Give each a probability: 0.25 good, 0.50 normal, 0.25 bad. They must add up to 1. Then multiply each outcome by its probability and add: that’s the **expected value**. $$ EV = \sum_i p_i \cdot x_i $$ *x_i: the outcome of scenario i; p_i: its probability. The p_i add up to 1.* **Alex Tutor:** 0.25 × 300,000 + 0.50 × 120,000 + 0.25 × (−80,000) = 75,000 + 60,000 − 20,000 = **115,000 UAH**. ![Each scenario’s profit times its probability gives 75,000, 60,000 and −20,000; their sum, 115,000 UAH, is the expected value.](https://productovi.com/courses/math-basics/diagrams/m7-l3-scenario-fork.png) *Weigh, then add.* **Sofia:** But no summer will actually bring 115,000. **Alex Tutor:** Right. EV is the average you’d get over many bets like this one. This summer will be one of the three outcomes, and there’s a 25% chance of losing 80,000. Always show the range next to the EV. > **Exercise:** A Club price test has three possible results for next quarter’s profit change: +90,000 UAH with probability 0.2, +30,000 with 0.6, −40,000 with 0.2. What is the expected change, in UAH? **Alex Tutor:** And one statement to check about that 115,000. > **Exercise:** Hanna’s scenarios for the summer intensive are +300,000 UAH, +120,000 and −80,000, with probabilities 0.25, 0.50 and 0.25, so the expected profit is 115,000 UAH. That makes about 115,000 UAH the most likely result of this summer. ## A pipeline of deals **Alex Tutor:** Klasna for Teams uses the same idea for its sales pipeline. A deal is worth win probability × lessons a year × contribution per lesson. One logistics company: 0.4 × 600 × 339 = 81,360 UAH. **Sofia:** And the whole pipeline is just the sum of those? **Alex Tutor:** Yes, EV adds up deal by deal. Just remember a 0.4 deal never brings 0.4 of the money: it brings all of it or nothing. **Sofia:** So ten small deals land closer to their EV than one big deal with the same EV? **Alex Tutor:** Usually, yes. Many independent bets average out: some win, some lose. One big deal at 0.4 is all or nothing, so a pipeline that leans on it is far riskier, even with the same expected value. > **Exercise:** Two deals: A has a 0.5 chance of 400 lessons a year, B a 0.25 chance of 1,200 lessons. Each lesson leaves 339 UAH of contribution. What is the pipeline’s expected contribution a year, in UAH? **Alex Tutor:** Here are two pipelines with the same expected value. > **Exercise:** Two Klasna for Teams pipelines. A: ten independent deals, each a 0.4 chance of 25,000 UAH a year. B: one deal, a 0.4 chance of 250,000 UAH a year. Which statement is right? ## A price that depends on the future **Alex Tutor:** Deals can include a payment that may or may not happen. Say Klasna buys a small tutoring platform for 4 million UAH now, plus 2 million more if the platform hits next year’s revenue target. That second payment is an **earn-out**. **Sofia:** So the price is either 4 or 6 million, depending on next year? **Alex Tutor:** In the end, yes. Today, though, you can price it: the sure part plus the uncertain part times its probability. > **Exercise:** Klasna buys a small tutoring platform for 4 million UAH now, plus 2 million more if the platform hits next year's revenue target. Klasna thinks the target has a 50% chance. What is the expected total price? ![A cat holding a small balance scale sits where three paths meet, with a full bowl, a half-full bowl and an empty bowl at their ends.](https://productovi.com/courses/math-basics/inline/m7-l3-i1.webp) *Every bowl counts, each in proportion to how likely it is.* ## Expected loss and insurance **Alex Tutor:** Expected value also explains insurance. Klasna’s recording studio kit is worth 200,000 UAH, and there’s about a 3% chance a year of losing it to a fire or theft. The **expected loss** is 0.03 × 200,000 = 6,000 UAH a year. **Sofia:** And the insurer asks 9,000. So insurance is a bad deal? **Alex Tutor:** On expected value it always is: the premium covers the insurer’s costs and profit. But without it, one bad year costs 200,000. With a 20,000 deductible the premium drops to 6,500: expected cost 6,500 + 0.03 × 20,000 = 7,100, worst case 26,500. ![No cover: 6,000 expected but up to 200,000; full cover: 9,000 for sure; a 20,000 deductible: 7,100 expected, at most 26,500.](https://productovi.com/courses/math-basics/diagrams/m7-l3-insurance-options.png) *Insure the losses you couldn’t absorb, not the small ones.* **Alex Tutor:** The deductible saves 2,500 on the premium and costs 20,000 only in a bad year. It pays off while the chance of a claim stays below 2,500 ÷ 20,000 = 12.5% a year. > **Exercise:** A tutor’s laptop costs 40,000 UAH, and the chance of losing or breaking it beyond repair in a year is 5%. What is the expected loss a year, in UAH? **Alex Tutor:** Now price a deductible yourself. > **Exercise:** A tutor’s 40,000 UAH laptop: full cover costs 3,200 UAH a year; with a 10,000 UAH deductible it costs 2,400. Up to what yearly chance of a claim does the deductible pay off, in percent? ## Units must match **Alex Tutor:** Project teams use the same product for risks: exposure = probability × impact. A 20% chance of a 10-day delay is an exposure of 2 days. A 10% chance of a 50,000 UAH overspend is 5,000 UAH. > **Exercise:** A project has a 20% chance of a 10-day delay (exposure 2 days) and a 10% chance of a 50,000 UAH overspend (exposure 5,000 UAH). From those two risks, the project’s total exposure is 2 + 5,000 = 5,002. ## Your turn **Alex Tutor:** At the planning meeting, Leo reads the summary and says: "Expected value 115,000, great, so the intensive makes 115,000." > **Exercise:** Hanna’s summer intensive has three scenarios: +300,000 UAH, +120,000 and −80,000, with probabilities 0.25, 0.50 and 0.25, so its expected profit is 115,000 UAH. In two or three sentences, explain to Leo, Klasna's CEO, what the 115,000 means and what he should look at next to it. ## Where you will meet this > **In short** > > - **Expected value** EV = Σ p·x: multiply each outcome by its probability and add. The p add up to 1. > - EV is an average over many bets; one bet lands on one outcome. Show the range too. > - Pipelines: win probability × lessons × margin, added deal by deal. Earn-out: sure part + p × uncertain part. > - **Expected loss** = p × L. A premium is always above it; insure the losses you can’t absorb. > - **Risk exposure** = probability × impact, only added within one unit. > **These lessons use it** > > - [The deal](/courses/bootstrapped-product/the-deal): valuing an earn-out you don’t control. > - [Insurance: big losses first](/courses/personal-finance/insurance-big-losses-first): a premium against the expected loss, and the deductible. > - [Case: Sam’s family protection checklist](/courses/personal-finance/protection-checklist-case): which risks to insure and which to carry. > - [Forecasting active users](/courses/product-analytics/forecasting): scenarios with a range instead of one number. > - [Scenarios and sensitivity](/courses/finance-for-product/scenarios-and-sensitivity): good, base and bad cases for a plan. **Alex Tutor:** Next time: many tests at once, and why twenty checks almost guarantee one false alarm. - Previous lesson: [Conditional probability and the 2×2 table](https://productovi.com/courses/math-basics/conditional-probability.md) - Next lesson: [Many tests at once](https://productovi.com/courses/math-basics/many-tests-at-once.md) --- Productovi Learning. 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