Ask how surprising the result is under a model
What does a small p-value actually say?
Entering midway? A hypothesis specifies a model before the data are evaluated. Independent binary trials with constant probability follow a binomial count distribution.
Start with a concrete question
Test whether an invented success probability is greater than a stated null value. Compute an exact upper-tail probability rather than assuming a normal approximation.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
Predict whether failing to reject proves the null is true.
Change one thing. Inspect what happens.
- At defaults inspect the probabilities for counts 8,9,10 and add them.
- Change observed successes while holding the null fixed.
- Change alpha and explain why choosing it after seeing the result undermines the planned test.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Exact binomial null distribution
Horizontal: Success count. Vertical: Probability.
- 1. Null probabilities (points)
Axes use the displayed ranges.
Read plotted values as a table
| Series | Success count | Probability |
|---|---|---|
| Null probabilities | 0 | 0.0009766 |
| Null probabilities | 1 | 0.009766 |
| Null probabilities | 2 | 0.04395 |
| Null probabilities | 3 | 0.1172 |
| Null probabilities | 4 | 0.2051 |
| Null probabilities | 5 | 0.2461 |
| Null probabilities | 6 | 0.2051 |
| Null probabilities | 7 | 0.1172 |
| Null probabilities | 8 | 0.04395 |
| Null probabilities | 9 | 0.009766 |
| Null probabilities | 10 | 0.0009766 |
Follow the calculation
- Null model: 10 independent Bernoulli trials with p=0.5; alternative: success probability greater than 0.5.
- Observed count 8; sum probabilities for counts 8 through 10.
- P(K ≥ 8 | null) = 0.0546875.
- Compare p-value 0.0546875 with alpha 0.05: do not reject this null in the specified test.
A p-value is a tail probability under the null model, not the probability that the null is true. Choose the alternative and threshold before observing data.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Ask how surprising the result is under a model
Recap for entering here
A hypothesis specifies a model before the data are evaluated. Independent binary trials with constant probability follow a binomial count distribution.
The concrete question
Test whether an invented success probability is greater than a stated null value. Compute an exact upper-tail probability rather than assuming a normal approximation.
Predict whether failing to reject proves the null is true.
Work through the mechanism
Under the null p=0.5 and n=10, counts at least 8 have probability (45+10+1)/1024=0.0546875.
At alpha=0.05 this result does not reject the null. It does not prove p=0.5; the data can be compatible with many probabilities.
The alternative is one-sided: greater success probability. Searching both directions or many outcomes after inspection needs a different procedure. A significant difference is not automatically a practically valuable effect.
Formula and symbols
P(K=k)=C(n,k)p₀ᵏ(1−p₀)ⁿ⁻ᵏ. Upper-tail p-value = Σ from k=observed to n. p₀ is the null probability; n is trial count.
Test the idea
At defaults inspect the probabilities for counts 8,9,10 and add them.
Change observed successes while holding the null fixed.
Change alpha and explain why choosing it after seeing the result undermines the planned test.
Assumptions and boundaries
Independent trials, fixed n and a constant success probability are assumed.
Repeated peeking, selective reporting and changing the tested hypothesis are not accounted for.
Extend your investigation
High school: Write a hypothesis and outcome rule before collecting any coin data.
College: Compare exact and approximate tests and simulate false-positive rates under fixed versus optional stopping.
Working professionals: Write an experiment decision rule combining effect size, uncertainty, cost and preselected evidence thresholds.
Try a fresh case
For four independent fair-coin trials, what is the one-sided probability of at least three heads?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Write a hypothesis and outcome rule before collecting any coin data.
College investigation
Compare exact and approximate tests and simulate false-positive rates under fixed versus optional stopping.
Workplace investigation
Write an experiment decision rule combining effect size, uncertainty, cost and preselected evidence thresholds.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.