Questions: Part 2 of 3 ere asked which ear they use to hear their cell phone, and the table is based on their responses. Determine whether a probability distribution is given. If a P(x) Left 0.6357 Right 0.3039 No preference 0.0604

Part 2 of 3 ere asked which ear they use to hear their cell phone, and the table is based on their responses. Determine whether a probability distribution is given. If a

P(x) Left 0.6357 Right 0.3039 No preference 0.0604

Transcript text: Part 2 of 3 ere asked which ear they use to hear their cell phone, and the table is based on their responses. Determine whether a probability distribution is given. If a \begin{tabular}{l|c|} \hline & $\mathbf{P}(\mathbf{x})$ \\ \hline Left & 0.6357 \\ \hline Right & 03039 \\ \hline \begin{tabular}{l} No \\ preference \end{tabular} & 0.0604 \\ \hline \end{tabular}

Solution

Solution Steps

To determine whether a probability distribution is given, we need to check two main conditions:

Each probability value must be between 0 and 1, inclusive.
The sum of all probability values must equal 1.

Solution Approach

Verify that each probability value is between 0 and 1.
Calculate the sum of all probability values and check if it equals 1.

Step 1: Verify Probability Values

We need to check if each probability value $ P(x) $ is within the range $ [0, 1] $. The given probabilities are:

$ P(\text{Left}) = 0.6357 $
$ P(\text{Right}) = 0.3039 $
$ P(\text{No preference}) = 0.0604 $

All values satisfy $ 0 \leq P(x) \leq 1 $:

$ 0 \leq 0.6357 \leq 1 $
$ 0 \leq 0.3039 \leq 1 $
$ 0 \leq 0.0604 \leq 1 $

Step 2: Calculate the Total Probability

Next, we calculate the sum of all probabilities: \[ \text{Total Probability} = P(\text{Left}) + P(\text{Right}) + P(\text{No preference}) = 0.6357 + 0.3039 + 0.0604 = 1.0 \]

Step 3: Determine Probability Distribution Validity

Since all individual probabilities are valid and the total probability equals 1, we conclude that the set of probabilities forms a valid probability distribution.