---
title: "Success/Failure Condition — AP Stats Definition & Guide"
description: "The success/failure (Large Counts) condition checks that expected successes and failures are at least 10, so p̂₁ − p̂₂ is approximately normal in AP Stats Unit 6."
canonical: "https://fiveable.me/ap-stats/key-terms/success-failure-condition"
type: "key-term"
subject: "AP Statistics"
unit: "Unit 3"
---

# Success/Failure Condition — AP Stats Definition & Guide

## Definition

The success/failure condition is the normality check for proportion inference: n₁p̂₁, n₁(1−p̂₁), n₂p̂₂, and n₂(1−p̂₂) must each be at least some set value (usually 10, sometimes 5) so the sampling distribution of p̂₁ − p̂₂ is approximately normal and a z-interval is valid.

## What It Is

The success/failure condition (your teacher may call it the **[Large Counts condition](/ap-stats/key-terms/large-counts-condition "fv-autolink")**) is the shape check you run before building a [two-sample z-interval](/ap-stats/unit-3/confidence-intervals-for-difference-two-proportions/study-guide/YjPeyk5dGYBwzENoOU6S "fv-autolink") for a difference in proportions. You verify that the number of successes and failures in *each* sample is big enough. In symbols, n₁p̂₁, n₁(1−p̂₁), n₂p̂₂, and n₂(1−p̂₂) must all be greater than or equal to some predetermined value, most commonly 10 (some textbooks use 5).

Why does this work? A sample proportion is secretly a binomial count divided by n, and binomial [distributions](/ap-stats/unit-1/describing-distribution-quantitative-variable/study-guide/4dcjgkWfLu7tmS9bDtjP "fv-autolink") only look bell-shaped when there are enough expected successes AND enough expected failures. If a sample has, say, only 3 successes, the distribution of p̂ is lopsided, the normal approximation breaks down, and your z* critical value no longer captures the confidence level you claim. Checking all four counts guarantees both sampling distributions are roughly normal, so their difference p̂₁ − p̂₂ is too.

## Why It Matters

This condition lives in **Topic 6.8 (Confidence Intervals for the Difference of Two Proportions)** in Unit 6: [Inference for Categorical Data: Proportions](/ap-stats/unit-3 "fv-autolink"). It directly supports learning objective **[AP Stats](/ap-stats "fv-autolink") 6.8.B**, which requires you to verify the conditions before calculating an interval. The CED splits the conditions into two jobs. Independence gets checked with random sampling and the 10% condition, and *shape* gets checked with success/failure. Skipping or botching this check is one of the most common ways to lose points on inference FRQs, because the normal model (and the z* you pull from it in 6.8.C) is only justified once this condition passes. The same logic echoes through every proportion procedure in Unit 6, so once you understand it here, you understand it everywhere.

## Connections

### [Independence Condition (Unit 6)](/ap-stats/key-terms/independence-condition)

These are the two halves of every condition check in Topic 6.8. [Independence](/ap-stats/key-terms/independence "fv-autolink") (random samples plus the 10% condition) justifies the standard error formula, while success/failure justifies the normal shape. You need both, and they answer different questions, so don't let one substitute for the other on an FRQ.

### Sampling Distribution of a Sample Proportion (Unit 5)

Success/failure isn't a new rule invented for inference. It's the exact same Large Counts condition from [Unit 5](/ap-stats/unit-5 "fv-autolink") that tells you when the sampling distribution of p̂ is approximately normal. Topic 6.8 just applies it twice, once per sample, because p̂₁ − p̂₂ inherits normality from its two pieces.

### Critical Value z* (Unit 6)

The z* in your interval formula comes from the [standard normal distribution](/ap-stats/key-terms/standard-normal-distribution "fv-autolink"). That's only legitimate if the sampling distribution actually is approximately normal, which is precisely what success/failure verifies. Pass the condition, earn the right to use z*.

### [Difference in Population Proportions (Unit 6)](/ap-stats/key-terms/difference-in-population-proportions)

The whole point of Topic 6.8 is estimating p₁ − p₂ with an interval. Success/failure is the gatekeeper that says your two-sample z-interval is a trustworthy way to do that estimation.

## On the AP Exam

On multiple choice, expect stems that hand you sample sizes and counts and ask what satisfying the condition lets you conclude. The answer is always about shape, specifically that the sampling distribution of p̂₁ − p̂₂ is approximately normal. For example, a question might say an analyst confirmed all four success and failure counts exceed 10 and ask what that allows the analyst to assume. On FRQs, two-proportion inference problems award credit for explicitly checking conditions with numbers, not just naming them. Write out all four products, like 'n₁p̂₁ = 48 ≥ 10,' and state the conclusion that the sampling distribution is approximately normal. A vague 'conditions are met' earns nothing. Also know the direction of logic. The condition justifies using the z-interval, so check it *before* computing, and if a count fails, say the normal approximation isn't appropriate rather than plowing ahead.

## success/failure condition vs 10% condition

Both are conditions you check in the same problem, but they do different jobs. The 10% condition (n₁ ≤ 10% of N₁ and n₂ ≤ 10% of N₂) is part of the independence check. It makes sure sampling without replacement doesn't mess up the standard error. The success/failure condition is the shape check that makes sure the sampling distribution is approximately normal. A sample can easily pass one and fail the other, so verify both separately and label which is which.

## Key Takeaways

- The success/failure (Large Counts) condition requires n₁p̂₁, n₁(1−p̂₁), n₂p̂₂, and n₂(1−p̂₂) to each be at least some predetermined value, usually 10, sometimes 5.
- Passing this condition lets you conclude the sampling distribution of p̂₁ − p̂₂ is approximately normal, which justifies using a z* critical value.
- It checks shape only; independence is a separate check handled by random sampling or random assignment plus the 10% condition.
- On FRQs, show the actual numbers for all four counts. Writing 'large counts met' without arithmetic does not earn the conditions point.
- For confidence intervals you check the condition using the sample proportions p̂₁ and p̂₂, since you have no hypothesized values to use instead.
- This is the same Large Counts idea from Unit 5 sampling distributions, applied to both samples at once in Topic 6.8.

## FAQs

### What is the success/failure condition in AP Stats?

It's the check that each sample has enough successes and failures, meaning n₁p̂₁, n₁(1−p̂₁), n₂p̂₂, and n₂(1−p̂₂) are all at least a set value (usually 10), so the sampling distribution of p̂₁ − p̂₂ is approximately normal and a two-sample z-interval is valid.

### Is the success/failure condition the same as the Large Counts condition?

Yes. Success/failure and Large Counts are two names for the same shape check. Some textbooks use a cutoff of 5 and others use 10, but the CED just says the counts must meet 'some predetermined value,' so use whichever cutoff your problem or teacher specifies and show the math.

### How is the success/failure condition different from the 10% condition?

The 10% condition (each sample is at most 10% of its population) is an independence check for sampling without replacement. Success/failure is a normality check on the counts of successes and failures. You need both before building a two-sample z-interval, and they are checked with completely different numbers.

### Does passing the success/failure condition mean my data is normal?

No. It means the *sampling distribution* of p̂₁ − p̂₂ is approximately normal. Your data is categorical (success or failure), so it can't be normal at all. The condition is about the distribution of the statistic across all possible samples, not about your raw data.

### What happens if the success/failure condition fails?

If any of the four counts falls below the cutoff, the normal approximation isn't trustworthy and a two-sample z-interval isn't appropriate. On the exam, say exactly that. State which count fails and conclude that the sampling distribution may not be approximately normal.

## Related Study Guides

- [3.10 Constructing a Confidence Interval for the Difference Between Two Population Proportions](/ap-stats/unit-3/confidence-intervals-for-difference-two-proportions/study-guide/YjPeyk5dGYBwzENoOU6S)

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