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 odd rolls in each outcome. for example, if the outcome is $ooe$, then $n(ooe) = 2$.

suppose that the random variable $x$ is defined in terms of $n$ as follows: $x = 2n - 2n^2 - 1$. the values of $x$ are given in the table below.

$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline \text{outcome} & eee & eoe & ooo & ooe & oee & oeo & eoo & eeo \\ \hline \text{value of } x & -1 & -1 & -13 & -5 & -1 & -5 & -5 & -1 \\ \hline \end{array}$

calculate the probabilities $p(x=x)$ 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.

Question

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 odd rolls in each outcome. for example, if the outcome is $ooe$, then $n(ooe) = 2$.

suppose that the random variable $x$ is defined in terms of $n$ as follows: $x = 2n - 2n^2 - 1$. the values of $x$ are given in the table below.

$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline \text{outcome} & eee & eoe & ooo & ooe & oee & oeo & eoo & eeo \\ \hline \text{value of } x & -1 & -1 & -13 & -5 & -1 & -5 & -5 & -1 \\ \hline \end{array}$

calculate the probabilities $p(x=x)$ 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.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Discrete Random Variable", "Probability Distribution" ], "new_concepts": [], "current_concepts": [ "Discrete Random Variable", "Probability Distribution" ] } ## Identify unique values of $X$ Using the *Discrete Random Variable* knowledge point $$ \begin{aligned} &\text{The values of } X \text{ given in the table are: } -1, -1, -13, -5, -1, -5, -5, -1.\ &\text{The unique values of } X \text{ in increasing order are: } -13, -5, -1. \end{aligned} $$ ## Calculate probabilities for each value of $X$ Using the *Probability Distribution* knowledge point $$ \begin{aligned} &\text{Total outcomes } = 8 \text{ (each equally likely with probability } \frac{1}{8}\text{).}\ &P(X = -13) = P(\{ooo\}) = \frac{1}{8}\ &P(X = -5) = P(\{ooe, oeo, eoo\}) = \frac{3}{8}\ &P(X = -1) = P(\{eee, eoe, oee, eeo\}) = \frac{4}{8} = \frac{1}{2} \end{aligned} $$ Calculate the probabilities $P(X=x)$ 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 $X$ | $-13$ | $-5$ | $-1$ | | :--- | :---: | :---: | :---: | | $P(X=x)$ | $\frac{1}{8}$ | $\frac{3}{8}$ | $\frac{1}{2}$ | { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Probability Distribution" ] }

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Question

Explanation:

Identify unique values of \(X\)

Calculate probabilities for each value of \(X\)

Answer: