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

Both Dice Show Even Numbers

What is an even number on a die?

Even numbers are divisible by 2: 2, 4, 6. That's 3 out of 6 faces. Odd numbers: 1, 3, 5 (also 3 out of 6 faces).

Why are two dice rolls independent?

Each die is a separate physical object. Rolling a 2 on the first die doesn't make the second die more or less likely to show an even number. They don't communicate!

Our approach:

Calculate P(first die even) and P(second die even) separately, then multiply because the events are independent.

Individual Probabilities

First Die: Even

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Second Die: Even

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Step-by-Step Solution

Find P(first die is even)

Even faces: 2, 4, 6 = 3 faces out of 6 total

Find P(second die is even)

Same calculation — the second die also has 3 even faces out of 6

Multiply (since events are independent)

Result:

Verification by Counting

Favorable outcomes: (2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6)

That is 3 × 3 = 9 outcomes out of 36 total.