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

Rolling Two Specific Numbers

What does "independent" mean?

Two events are independent if knowing one happened gives you no information about whether the other will happen.

For dice: The first die doesn't "know" what it rolled. The second die doesn't "care" what the first one did. Each roll is a fresh start.

Why can we multiply for independent events?

If A and B are independent, then P(B | A) = P(B). The general formula:

becomes simpler when independent:

The key test for independence:

Ask yourself: "Does the outcome of the first event change the probability of the second?" For separate dice rolls, the answer is no — so they're independent.

Example: Rolling 6, Then Another 6

First Roll: 6

✓

Second Roll: 6

✓

Apply multiplication rule

Result:

Example: Rolling 3, Then 5

First Roll: 3

✓

Second Roll: 5

✓

Result:

Key Insight

Any specific sequence of two numbers has probability 1/36. This is because there are 6 × 6 = 36 equally likely outcomes when rolling two dice.