Let's break down the problem step-by-step:

Problem Summary

1. Experiment: A fair die is rolled 3 times.
2. Outcome Representation: Each roll can be categorized as odd (o) or even (e).
   • An outcome, such as "oee," indicates odd on the first roll, even on the second, and even on the third.
3. Random Variable $N$: Counts the number of even rolls in each outcome.
   • Example: For "eoe," $N = 2$ since there are two even numbers.
4. Random Variable $X$: Defined in terms of $N$ as: $X = -2N^2 - 4N + 3$
   • This equation determines the value of $X$ for each outcome.

Calculating Values of $X$

Let's calculate $N$ and then $X$ for each outcome:

1. Outcome: eee
   • $N = 3$ (3 even rolls)
   • $X = -2(3)^2 - 4(3) + 3 = -18 - 12 + 3 = -27 + 3 = -24$
2. Outcome: oee
   • $N = 2$
   • $X = -2(2)^2 - 4(2) + 3 = -8 - 8 + 3 = -16 + 3 = -13$
3. Outcome: eoe
   • $N = 2$
   • $X = -2(2)^2 - 4(2) + 3 = -8 - 8 + 3 = -16 + 3 = -13$
4. Outcome: eoo
   • $N = 1$
   • $X = -2(1)^2 - 4(1) + 3 = -2 - 4 + 3 = -3$
5. Outcome: oeo
   • $N = 1$
   • $X = -2(1)^2 - 4(1) + 3 = -2 - 4 + 3 = -3$
6. Outcome: ooe
   • $N = 1$
   • $X = -2(1)^2 - 4(1) + 3 = -2 - 4 + 3 = -3$
7. Outcome: eoo
   • $N = 1$
   • $X = -2(1)^2 - 4(1) + 3 = -2 - 4 + 3 = -3$

Probability Distribution of $X$

To calculate the probability distribution, we need to consider:

1. Total number of outcomes: There are $2 \times 2 \times 2 = 8$ outcomes (since each roll can be either odd or even).
2. Probability of each outcome: Since the die is fair, each outcome has a probability of $\frac{1}{8}$.

Now, let's tabulate the probability of each value of $X$:

Value of $X$	Occurrences	Probability $P(X = x)$
-24	1	$\frac{1}{8}$
-13	2	$\frac{2}{8} = \frac{1}{4}$
-3	5	$\frac{5}{8}$

Final Answer

The probability distribution is:

• $P(X = -24) = \frac{1}{8}$
• $P(X = -13) = \frac{1}{4}$
• $P(X = -3) = \frac{5}{8}$

Would you like a further breakdown or additional help on this problem?

Related Questions:

1. What is the probability of having exactly one even roll?
2. How does changing the random variable formula affect the probability distribution?
3. What is the expectation value (mean) of $X$?
4. How many different outcomes are there if we consider $N$ alone?
5. What is the variance of the random variable $X$?

Tip: When solving probability problems, always count the total outcomes and ensure that the sum of probabilities equals 1.