Bayes' Theorem with Dice

Update beliefs about dice based on evidence

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Loaded Die DetectionLoaded Fair Given ResultFair?Which Die?Which?

Is the Die Loaded?

What is Bayes' Theorem?

Bayes' theorem tells us how to update our beliefs when we observe new evidence. The formula is:

Where H is our hypothesis (die is loaded) and E is the evidence (three 6s).

The key idea:

We start with a prior belief (how likely we think the die is loaded before testing), observe evidence, then calculate a posterior belief (how likely after seeing the evidence).

The Scenario

You suspect a die might be loaded (biased toward 6). You roll it three times and get three 6s in a row. What is the probability that the die is actually loaded?

Setting Up the Problem

Prior Probability

Before seeing any evidence, assume 10% of dice are loaded:

Die Probabilities

For rolling a 6:

(assumed)

Step-by-Step Solution

Calculate P(three 6s | loaded)

If the die is loaded (P(6) = 0.5), getting three 6s:

Calculate P(three 6s | fair)

If the die is fair (P(6) = 1/6), getting three 6s:

Calculate P(three 6s) - Total Probability

Using the law of total probability:

Apply Bayes' Theorem

Answer

After observing three 6s in a row, the probability that the die is loaded increased from 10% to approximately 75%.

Prior (before evidence)

10%

Posterior (after evidence)

75%

Key Takeaways

1.Three 6s is much more likely with a loaded die (12.5%) than a fair die (0.46%)
2.The evidence strongly shifts our belief toward "loaded"
3.But we are not 100% certain - a fair die can still roll three 6s by chance