Math Problem Statement
Solve for the value of k and find P(X=0) based on the probability distribution table.
Solution
The question involves solving for the value of and finding . The given data appears to represent a probability distribution table:
\hline x & 0 & 1 & 2 & 3 \\ \hline P(X=x) & 0.1 & k & 2k & 0.2 \\ \hline \end{array}$$ The total probability of all outcomes must equal 1. Thus: $$P(X=0) + P(X=1) + P(X=2) + P(X=3) = 1$$ Substituting the values: $$0.1 + k + 2k + 0.2 = 1$$ ### Step 1: Solve for $$k$$ Simplify the equation: $$0.1 + 0.2 + 3k = 1$$ $$0.3 + 3k = 1$$ $$3k = 1 - 0.3$$ $$3k = 0.7$$ $$k = \frac{0.7}{3} = 0.2333\ (\text{approximately})$$ ### Step 2: Find $$P(X=0)$$ From the table, $$P(X=0) = 0.1$$. ### Final Results: - $$k = 0.2333$$ (approximately) - $$P(X=0) = 0.1$$ Would you like a detailed explanation or any additional help with this? 😊 ### Expanding on This: 1. What does a probability distribution mean in statistics? 2. How do we ensure that a given table satisfies the properties of a probability distribution? 3. What is the interpretation of $$P(X=0)$$? 4. How can you verify your solution for $$k$$ by substitution? 5. What happens if one of the probabilities exceeds 1? ### Tip: When working with probabilities, always check that the sum equals 1 to confirm it's a valid distribution!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability Distribution
Algebra
Linear Equations
Formulas
Sum of probabilities: P(X=0) + P(X=1) + P(X=2) + P(X=3) = 1
Theorems
Sum of probabilities theorem
Suitable Grade Level
Grades 9-12