To solve this problem, we need to calculate the values of the random variable $X$ based on the given outcomes and then determine the probability distribution of $X$. Let's break it down step by step.

Step 1: Understanding the Problem

• We are rolling a fair die (which has numbers from 1 to 6) three times.
• For each roll, the outcome can be either even (E) or odd (O).
• Each outcome can be represented as a string like "EEE" or "OEO", where E stands for an even result and O stands for an odd result.
• We are interested in the random variable $N$, which counts the number of even results in each outcome.

Given that the random variable $X$ is defined as: $X = 6N - 2N - 1$ we first need to calculate the value of $X$ for each possible outcome.

Step 2: Calculate $X$ for All Possible Outcomes

• There are $2^3 = 8$ possible outcomes for three rolls since each roll can result in either an even or an odd number.

Let's list the possible outcomes and calculate $N$ and $X$ for each:

Step 3: Determine the Probability Distribution of $X$

Now we need to calculate the probabilities for each unique value of $X$. Since the die is fair and each outcome has an equal probability of $\frac{1}{8}$, we just need to count how many times each value of $X$ occurs:

• $X = -1$ occurs 1 time (OOO)
• $X = 3$ occurs 3 times (OOE, OEO, EOO)
• $X = 7$ occurs 3 times (OEE, EOE, EEO)
• $X = 11$ occurs 1 time (EEE)

So the probability distribution of $X$ is: