Business Context
At PlayGrid, a mobile gaming company, a simple two-dice mechanic is used in a rewards feature. A product analyst wants to quantify the chance of a specific outcome after partial information is revealed to the player.
Problem Statement
Two fair six-sided dice are rolled. You are told that the first die shows 5. Given that information, determine the probability that the sum of the two dice is 8.
Given Data
| Variable | Value |
|---|
| Number of dice | 2 |
| Sides per die | 6 |
| First die observed | 5 |
| Target sum | 8 |
| Die fairness assumption | Uniform over $1 to \6$ |
Requirements
- Define the conditional probability event clearly.
- List the sample space after conditioning on the first die being 5.
- Count the favorable outcomes where the total is 8.
- Compute the conditional probability using the formal formula.
- Briefly compare this result with the unconditional probability that two dice sum to 8.
Assumptions
- Both dice are fair and independent.
- Each die has outcomes {1,2,3,4,5,6} with equal probability.
- The first die being 5 is known with certainty before evaluating the target event.
- Order matters because the dice are distinguishable as first and second die.
