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Random Variables and Distributions
Discrete probability distribution: Word problems
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An number cube (a fair die) is rolled 3 times. For each roll, we are interested in whether the roll comes up even or odd. An outcome is represented by a string of the sort oee (meaning an odd number on the first roll, an even number on the second roll, and an even number on the third roll).
For each outcome, let N be the random variable counting the number of even rolls in each outcome. For example, if the outcome is eoe, then =Neoe2. Suppose that the random variable X is defined in terms of N as follows: =X−2N2−4N3. The values of X are given in the table below.
Outcome	eeo	ooe	oee	eee	eoe	eoo	oeo	ooo
Value of X	−3	−5	−3	3	−3	−5	−5	−3
Calculate the probabilities P=Xx of the probability distribution of X. First, fill in the first row with the values of X. Then fill in the appropriate probabilities in the second row.
Value x of X
P=Xx

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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.

An number cube (a fair die) is rolled 3 times. For each roll, we are interested in whether the roll comes up even or odd. An outcome is represented by a string of the sort oee (meaning an odd number on the first roll, an even number on the second roll, and an even number on the third roll). For each outcome, let N be the random variable counting the number of even rolls in each outcome. Suppose that the random variable X is defined in terms of N as follows: X = -2N^2 - 4N + 3. Calculate the probabilities P(X=x) of the probability distribution of X.

Learn how to calculate the probability distribution of a random variable X derived from rolling a fair die 3 times. This problem explores combinatorics, probability, and the formula X = -2N^2 - 4N + 3, which counts even rolls. Suitable for Grades 10-12.

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