Conditional Dice Probability

P(B|A) with dice examples

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P(Roll 6 | Rolled an Even Number)

What does P(A | B) mean?

The "|" symbol means "given that". So P(6 | even) reads as: "The probability of rolling a 6, given that we already know the roll was even."

How does "given" change things?

We restrict our view to only the outcomes where the condition is true. Instead of considering all 6 die faces, we only consider the even faces (2, 4, 6). This is called the reduced sample space.

Question

If you rolled a die and were told the result is an even number, what is the probability that it was a 6?

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Step-by-Step Solution

Define the Events

Let A = "rolled an even number"

Let B = "rolled a 6"

We want to find P(B|A) = P(6 | even)

Identify the Reduced Sample Space

Given A occurred (even roll), our new sample space is:

Count Favorable Outcomes

Within the even numbers, how many are 6?

Apply the Conditional Probability Formula

Or simply: 1 favorable out of 3 possible even outcomes = 1/3

Answer

Key Insight

Notice that P(6) = 1/6 unconditionally, but P(6|even) = 1/3. The conditional probability is higher because we've "zoomed in" on a smaller sample space where 6 makes up a larger proportion!