Statistics

Is this coin fair?

Make a conjecture. Compare it with chance.

Predict, test and explain

Step 1 of 3

Predict

At a 5% significance level, what would a p-value of 0.03 tell you to do?

Make a prediction, then toss.

Assume the coin is fair\text{Assume the coin is fair}
If this coin were fair,\text{If this coin were fair,}
what results would you expect?\text{what results would you expect?}
P(heads)=12P(\text{heads})=\frac12

Scroll the diagram sideways to see every label.

Collect one sample, then compare it with fair coins.

InvestigateWould your result be unusual if this coin were fair?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

A pre-planned test of a fair coin gives p = 0.03 at a 5% significance level. What is the decision?

p=0.03,α=0.05p=0.03,\quad \alpha=0.05

Hint

Compare the p-value with the significance level.

Reveal answer

Reject the fair-coin hypothesis at 5%. There is evidence against fairness; this does not prove the coin is biased.

0.03<0.050.03<0.05

Build an explanation

Hint 1

Predict results assuming the coin is fair before collecting a sample.

Hint 2

Compare your result with fair-coin experiments of the same size.

Hint 3

Unusual results provide evidence against the fair-coin model, but a single experiment cannot prove fairness or bias.

Worked example

This example uses fixed values, separate from the diagram controls.

In 10 independent tosses, a coin gives 10 heads. Is that unusual under a fair-coin model?

  1. A fair coin has 2¹⁰ = 1024 equally likely sequences.
  2. All heads and all tails are equally extreme: 2/1024 = 1/512.
  3. This is about 0.195%, below a prechosen 5% level.

Answer: Reject at 5%; this does not prove bias.

Connect this idea

Watch out: A p-value measures how unusual results are under the model, not the probability the coin is fair.