Math Problem Statement

For families with 5 children, let X be the number of children with Genetic Condition B. Can the following table be a probability distribution for the random variable X?

Solution

To determine if the given table represents a valid probability distribution for the random variable XX, two conditions must be met:

  1. Each probability must be between 0 and 1: This ensures that the values are valid probabilities.
  2. The sum of all probabilities must equal 1: This ensures that the total probability across all outcomes is correctly normalized.

Given the table:

\hline x & P(x) \\ \hline 1 & 0.2869 \\ 2 & 0.0705 \\ 3 & 0.2241 \\ 4 & 0.192 \\ 5 & 0.2265 \\ \hline \end{array}$$ ### Step 1: Check if each probability is valid. All probabilities $$P(x)$$ are between 0 and 1, so this condition is satisfied. ### Step 2: Check if the sum of probabilities equals 1. Let's sum the probabilities: $$0.2869 + 0.0705 + 0.2241 + 0.192 + 0.2265 = 0.9999999999999999$$ This is extremely close to 1 (due to rounding error), so this condition is also satisfied. ### Conclusion: The table represents a valid probability distribution. The answer to the question is **"yes"**. Would you like any further details or explanations on this? --- Here are five related questions for further exploration: 1. What is the expected value of a discrete random variable? 2. How do you calculate the variance of a probability distribution? 3. What is the difference between a probability distribution and a cumulative distribution function (CDF)? 4. Can a probability distribution have negative values for probabilities? 5. How would the distribution change if another value of $$x$$ was added? **Tip:** When dealing with probabilities, always ensure the sum of all possible outcomes equals 1!

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Math Problem Analysis

Mathematical Concepts

Probability Distribution
Discrete Random Variables

Formulas

P(X) = 1 (Sum of probabilities)
0 ≤ P(x) ≤ 1 (Probability constraint)

Theorems

Probability Distribution Theorem
Sum of Probabilities = 1

Suitable Grade Level

Grades 9-12

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