Happiness In a survey sponsored by Coca-Cola, subjects were asked what contributes most to their happiness, and the table summarizes their responses. Does the table represent a probability distribution? Explain.

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Step 1: Understand the requirements for a probability distribution. A probability distribution must satisfy two conditions: (1) The sum of all probabilities must equal 1, and (2) each individual probability must be between 0 and 1, inclusive.

Step 2: Examine the table provided. The table lists the categories (Family/partner, Friends, Other) and their corresponding probabilities (P(x)): 0.77, 0.15, and 0.08.

Step 3: Verify condition (1). Add the probabilities together: \( P(x) = 0.77 + 0.15 + 0.08 \). Check if the sum equals 1.

Step 4: Verify condition (2). Check each probability value to ensure it is between 0 and 1. Specifically, confirm that \( 0 \leq 0.77 \leq 1 \), \( 0 \leq 0.15 \leq 1 \), and \( 0 \leq 0.08 \leq 1 \).

Step 5: Based on the results of steps 3 and 4, determine whether the table represents a valid probability distribution. If both conditions are satisfied, the table is a probability distribution; otherwise, it is not.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability Distribution

A probability distribution describes how the probabilities of a random variable are distributed across its possible values. For a valid probability distribution, the sum of all probabilities must equal 1, and each individual probability must be between 0 and 1. In the context of the survey, the table lists the probabilities associated with different contributors to happiness.

Total Probability

Total probability refers to the sum of all probabilities in a probability distribution. It is a crucial aspect to verify if a set of probabilities forms a valid distribution. In the given table, the total probability is calculated by adding the probabilities of Family/partner (0.77), Friends (0.15), and Other (0.08), which equals 1.00, confirming that it meets the criteria for a probability distribution.

Random Variable

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. In this case, the random variable represents different contributors to happiness, such as Family/partner, Friends, and Other. Each contributor has an associated probability, indicating the likelihood of it being a significant factor in the respondents' happiness.

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