Independent Events: Dice

P(A ∩ B) = P(A) × P(B)

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Two Specific RollsTwo Rolls Both Dice EvenBoth Even First Roll × Second RollProduct

Multiple Rolls: Extended Example

Can we extend the multiplication rule beyond 2 events?

Yes! For any number of independent events, we just keep multiplying. If A, B, and C are all independent:

Why does this work?

Each roll is independent of all previous rolls. The die has no memory. Even if you've rolled 10 sixes in a row (incredibly rare!), the next roll still has a 1/6 chance of being a six.

This is called the "Gambler's Fallacy" to believe otherwise!

The general formula:

Example: Rolling Three 6s in a Row

Roll 1

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Roll 2

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Roll 3

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Result:

Probability of Rolling All 6s

Number of Rolls	Formula	Probability
1
2
3
4
n

Why This Works

Independence means past outcomes do not influence future ones. Even after rolling five 6s in a row, the probability of the next roll being a 6 is still 1/6. The dice have no memory!