Math Problem Statement
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
oeo
, then
=Noeo2
. Suppose that the random variable
X
is defined in terms of
N
as follows:
=X−2N2−4N1
. The values of
X
are given in the table below.
Outcome
eee
ooo
eeo
eoo
ooe
oeo
eoe
oee
Value of
X
−1
5
−3
−1
−1
−1
−3
−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
Solution
To solve this problem, we need to first calculate the probabilities associated with each possible value of the random variable based on the given outcomes and the number of odd rolls .
Step 1: Analyze the Outcomes and Calculate for Each Outcome
Given the outcomes and the values of :
\hline \text{Outcome} & N & X \\ \hline \text{eee} & 0 & -1 \\ \text{ooo} & 3 & 5 \\ \text{eeo} & 1 & -3 \\ \text{eoo} & 2 & -1 \\ \text{ooe} & 2 & -1 \\ \text{oeo} & 2 & -1 \\ \text{eoe} & 1 & -3 \\ \text{oee} & 1 & -3 \\ \hline \end{array}$$ ### Step 2: Count the Frequency of Each Value of $$X$$ Now, we count how many times each value of $$X$$ appears among the outcomes: $$\begin{array}{|c|c|} \hline \text{Value of } X & \text{Frequency} \\ \hline -1 & 4 \\ 5 & 1 \\ -3 & 3 \\ \hline \end{array}$$ ### Step 3: Calculate the Probability $$P(X = x)$$ Since each outcome is equally likely, and there are 8 possible outcomes (since each roll can result in either an even or odd number and there are 3 rolls), the probability of each outcome is $$\frac{1}{8}$$. The probability of each value of $$X$$ is then calculated by multiplying the frequency by $$\frac{1}{8}$$. $$\begin{array}{|c|c|} \hline \text{Value of } X & P(X = x) \\ \hline -1 & \frac{4}{8} = 0.50 \\ 5 & \frac{1}{8} = 0.125 \\ -3 & \frac{3}{8} = 0.375 \\ \hline \end{array}$$ ### Final Answer: $$\begin{array}{|c|c|c|c|} \hline \text{Value of } X & -1 & 5 & -3 \\ \hline P(X = x) & 0.50 & 0.125 & 0.375 \\ \hline \end{array}$$ Would you like more details or further clarifications? Here are 5 related questions you might find interesting: 1. How would the probabilities change if the number of rolls increased to 4? 2. What is the expected value of the random variable $$X$$? 3. How do you calculate the variance of $$X$$? 4. What if we were interested in whether the rolls were prime numbers instead of odd/even—how would that change the problem? 5. How can you verify that the total probability $$P(X = x)$$ sums to 1? **Tip:** When working with probability distributions, always check that the sum of the probabilities equals 1 to ensure correctness.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Random Variables
Probability Distributions
Formulas
-
Theorems
-
Suitable Grade Level
Advanced High School