Question

A pair of dice is thrown simultaneously. If \( X \) denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of \( X \).

Hint:

For probability distributions, count outcomes carefully and verify the total probability sums to 1.

Answer 1

Step 1: Define the values of \( X \).
The possible absolute differences are: \[ X = 0, 1, 2, 3, 4, 5. \] Step 2: Count the outcomes for each \( X \).
- For \( X = 0 \): Both dice show the same number. There are 6 outcomes: \[ (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6). \] - For \( X = 1 \): The numbers differ by 1. Outcomes are: \[ (1, 2), (2, 1), (2, 3), (3, 2), \ldots, (5, 6), (6, 5). \] This gives \( 2 \times 5 = 10 \) outcomes. - Repeat similar counting for \( X = 2, 3, 4, 5 \). Step 3: Calculate probabilities.
The total number of outcomes is \( 6 \times 6 = 36 \). Probabilities are: \[ P(X = k) = \frac{\text{Number of favorable outcomes for } X = k}{36}. \] Step 4: Write the probability distribution.
\[ P(X = 0) = \frac{6}{36}, \quad P(X = 1) = \frac{10}{36}, \quad P(X = 2) = \frac{8}{36}, \ldots \]