# p and 1 − p

> Probability as a share, its complement 1 − p, retention as 1 − churn, and why p(1 − p) peaks at 0.5.

- Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md)
- Module: Probability and distributions
- Lesson 25/36, ~26 min
- Source: https://productovi.com/courses/math-basics/p-and-1-minus-p

## A probability is a share

**Alex Tutor:** A new module: probability. At Klasna it hides in plain sight. In September, 400 people took a free trial lesson and 120 of them bought a package afterwards.

**Sofia:** So the chance that a trial learner buys is 120 out of 400?

**Alex Tutor:** Yes: p = 120 ÷ 400 = 0.30. A probability is a share of many similar cases, always between 0 (never) and 1 (always). For the next trial learner we can't say yes or no, but we can say 0.30.

**Alex Tutor:** Try it on October’s numbers before we go on.

> **Exercise:** In October, 150 people took a free trial lesson at Klasna and 36 of them bought a package. Which reading of these numbers is right?

**Sofia:** And the other 280?

**Alex Tutor:** They are the **complement**: everyone who didn't buy. Its probability is 1 − p = 1 − 0.30 = 0.70. "Buys" and "doesn't buy" cover every case and never overlap, so the two shares must add up to 1.

$$
P(\text{not } A) = 1 - P(A)
$$

*The complement rule. P(A): the probability that A happens.*

> **Exercise:** Match each event at Klasna with its complement.

**Alex Tutor:** One check before we use the rule anywhere.

> **Exercise:** In September, 30% of Klasna’s trial learners bought a package and 25% bought a single lesson. So "buys a single lesson" is the complement of "buys a package", and the two shares must add up to 1.

## Retention is 1 − churn

**Alex Tutor:** You already know one complement from subscriptions. Klasna Club loses 8% of its members every month: that's monthly **churn**. The share that stays is the **retention**: 1 − 0.08 = 0.92.

![Of 400 trial learners, 0.30 bought and 0.70 did not; in Klasna Club 92% stay and 8% cancel. The two parts always add up to 1.](https://productovi.com/courses/math-basics/diagrams/m7-l1-complement-bar.png)

*One part tells you the other: p and 1 − p, retention and churn.*

**Sofia:** So a report that shows only churn also shows retention.

**Alex Tutor:** Exactly. And the complement is often the easier side to count. "At least one lesson missed this month" is hard to count directly; "no lesson missed" is one number, and 1 minus it gives the rest.

> **Exercise:** In March, Klasna Club’s monthly churn was 6.5%. What was the monthly retention, in percent?

## Why p(1 − p) keeps coming back

**Sofia:** In the statistics course I saw p(1 − p) inside a square root. Why multiply a probability by its own complement?

**Alex Tutor:** Code each trial learner as 1 if they bought and 0 if not. The spread of those zeros and ones, the variance, works out to exactly p(1 − p). It measures how hard the next answer is to guess.

**Alex Tutor:** Try a few values. p = 0.1 gives 0.1 × 0.9 = 0.09. p = 0.3 gives 0.21. p = 0.5 gives 0.25. Then it falls again: 0.7 gives 0.21, 0.9 gives 0.09.

![p(1 − p) is the area of a rectangle with sides p and 1 − p: 0.09 at 0.1, 0.21 at 0.3, a peak of 0.25 at 0.5, then the same values in mirror.](https://productovi.com/courses/math-basics/diagrams/m7-l1-pq-peak.png)

*Two sides that always add up to 1 give the biggest area as a square.*

**Sofia:** It peaks in the middle, and 0.2 gives the same as 0.8.

**Alex Tutor:** Right, swapping p and 1 − p changes nothing. Picture a rectangle with sides p and 1 − p: the sides always add up to 1, and a square holds the most area. At 0.5 the answer is a coin toss; at 0.02 you’d guess "no" and almost always be right.

> **Exercise:** Four Klasna funnels have these conversion rates. For the same number of people, which one has the largest p(1 − p)?

## Where it shows up

**Alex Tutor:** The standard error of a share is SE = √(p(1 − p) ÷ n). For our trial: √(0.21 ÷ 400) = √0.000525 ≈ 0.023, so the 30% is known to about ±2.3 pp per standard error.

$$
SE_p = \sqrt{\frac{p(1-p)}{n}}
$$

*p: the observed share; n: how many people it was measured on.*

**Sofia:** And if we haven’t measured p yet, for a new survey?

**Alex Tutor:** Plan with p = 0.5. It gives the largest p(1 − p), so the largest SE: √(0.25 ÷ 400) = 0.025. Whatever the true share turns out to be, the error can only be smaller, never bigger.

> **Exercise:** A sample-size formula for an A/B test contains p₁(1 − p₁). The baseline conversion is p₁ = 0.04. What is p₁(1 − p₁)? Give four decimals.

**Alex Tutor:** That one comes straight from the product-analytics course: its sample-size formula adds p₁(1 − p₁) + p₂(1 − p₂). With 4% and 5% that is 0.0384 + 0.0475 = 0.0859.

> **Exercise:** For a share of 20% and a share of 80%, measured on the same number of people, the standard error √(p(1 − p) ÷ n) is the same.

**Alex Tutor:** Last check before your turn: the planning case.

> **Exercise:** Hanna, Klasna's head of finance, wants to survey 625 Klasna parents about a share nobody has measured yet. Planning with p = 0.5, what is the largest standard error √(p(1 − p) ÷ n) she should expect, in percentage points?

## Your turn

**Alex Tutor:** Hanna is planning a survey of Klasna parents: what share would pay for a Club subscription for their child? Nobody knows the rate yet, and she sized the sample with p = 0.5.

> **Exercise:** A colleague asks Hanna why she used p = 0.5 when "surely far fewer than half will say yes". In two or three sentences, explain why p = 0.5 is the safe choice.

## Where you will meet this

> **In short**
>
> - A **probability** is a share of many similar cases, from 0 to 1.
> - The **complement**: P(not A) = 1 − P(A). Retention = 1 − churn.
> - **p(1 − p)** is the spread of a yes/no outcome. It is largest at p = 0.5 (0.25) and the same for p and 1 − p.
> - So SE = √(p(1 − p) ÷ n) is largest at 0.5: plan with 0.5 when the share is unknown.

> **These lessons use it**
>
> - [95% confidence intervals](/courses/statistics-for-pm-and-marketers/confidence-intervals): the standard error of a share, √(p(1 − p)/n).
> - [Power analysis and sample size](/courses/statistics-for-pm-and-marketers/power-and-sample-size): why the sample is planned with a guess of p.
> - [Conversion rates and the two-proportion z-test](/courses/statistics-for-pm-and-marketers/conversion-rate-tests): p(1 − p) for two groups.
> - [How big and how long](/courses/product-analytics/test-size): p₁(1 − p₁) + p₂(1 − p₂) in the users-per-variant formula.
> - [MRR and churn](/courses/bootstrapped-product/mrr-and-churn): retention as 1 − churn.

**Alex Tutor:** Next time: probability with a condition, "given that", and the 2×2 table behind precision and recall.

- Previous lesson: [e, ln, odds and the sigmoid](https://productovi.com/courses/math-basics/e-ln-odds-sigmoid.md)
- Next lesson: [Conditional probability and the 2×2 table](https://productovi.com/courses/math-basics/conditional-probability.md)

---
Productovi Learning. Free, no sign-up needed to read.