Groups of adults are randomly selected and arranged in groups of three.
The random variable x is the number in the group who say that they would feel comfortable in a self-driving vehicle. Determine whether a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied.
x
0
1
2
3
P(x)
0.352
0.437
0.183
0.028
Does the table show a probability distribution? Select all that apply.
A. Yes, the table shows a probability distribution.
B. No, the random variable x's number values are not associated with probabilities.
C. No, the random variable x is categorical instead of numerical.
D. No, the sum of all the probabilities is not equal to 1.
• E. No, not every probability is between 0 and 1 inclusive.

To determine whether the table shows a probability distribution, we need to check if the following requirements are satisfied:

1. The sum of all probabilities must equal 1.
2. Each individual probability \(P(x)\) must be between 0 and 1 inclusive.
3. The random variable \(x\) must represent numerical values (i.e., it must be a quantitative variable).

### Given Data:

- \(x\): 0, 1, 2, 3
- \(P(x)\): 0.352, 0.437, 0.183, 0.028

### Step 1: Check if the probabilities sum to 1.
\[
P(0) + P(1) + P(2) + P(3) = 0.352 + 0.437 + 0.183 + 0.028 = 1.000
\]
The sum of the probabilities is exactly 1.

### Step 2: Check if each probability is between 0 and 1 inclusive.
\[
0.352, 0.437, 0.183, 0.028 \text{ are all between 0 and 1 inclusive.}
\]

### Step 3: Check if \(x\) is a numerical variable.
The random variable \(x\) represents the number of people who feel comfortable in a self-driving vehicle, which is a quantitative (numerical) variable.

### Conclusion:
- The table meets all the criteria for a probability distribution.

Thus, the correct answer is:
**A. Yes, the table shows a probability distribution.**

Would you like to calculate the mean and standard deviation of the probability distribution next? Here are some related questions to expand on this:

1. How do you calculate the mean of a probability distribution?
2. How do you calculate the variance and standard deviation for a probability distribution?
3. What is the interpretation of the mean in this context?
4. How does the standard deviation relate to the spread of the distribution?
5. How could the probability distribution change if the sample size increases?

**Tip:** When analyzing probability distributions, always verify that the probabilities add up to 1 as a first step.

Groups of adults are randomly selected and arranged in groups of three. The random variable x is the number in the group who say that they would feel comfortable in a self-driving vehicle. Determine whether a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied.

x
0
1
2
3
P(x)
0.352
0.437
0.183
0.028

Does the table show a probability distribution? Select all that apply.
A. Yes, the table shows a probability distribution.
B. No, the random variable x's number values are not associated with probabilities.
C. No, the random variable x is categorical instead of numerical.
D. No, the sum of all the probabilities is not equal to 1.
E. No, not every probability is between 0 and 1 inclusive.

Learn how to determine if a table of random variables and probabilities represents a valid probability distribution. This example checks conditions including the sum of probabilities and whether each probability is between 0 and 1. Suitable for students in Grades 10-12.

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