https://file.aiquickdraw.com/mathbot/file/1729122461160ae1rc4m9.jpg

The image presents a probability model with the number of automobiles owned by a randomly chosen family and their corresponding probabilities. The task is to calculate the mean number of automobiles owned.

### Table:

| Number of automobiles | 0   | 1   | 2   | 3   |
|-----------------------|-----|-----|-----|-----|
| Probability           | 0.15| 0.20| 0.55| 0.10|

### Formula for Mean:
The mean number of automobiles owned, denoted as \( \mu \), is calculated using the expected value formula for a discrete probability distribution:

\[
\mu = \sum (x_i \cdot P(x_i))
\]

Where:
- \(x_i\) is the number of automobiles, and
- \(P(x_i)\) is the corresponding probability.

### Calculation:
\[
\mu = (0 \cdot 0.15) + (1 \cdot 0.20) + (2 \cdot 0.55) + (3 \cdot 0.10)
\]
\[
\mu = (0) + (0.20) + (1.10) + (0.30)
\]
\[
\mu = 1.60
\]

So, the mean number of automobiles owned is **1.6**.

Let me know if you'd like more details or have any questions! Here are five related questions for further exploration:

1. How would the mean change if the probability of owning 3 automobiles increased?
2. What is the variance of this probability distribution?
3. How would you interpret a mean that is not a whole number?
4. Can you find the cumulative distribution for this data?
5. How would this model change if the probabilities were skewed towards owning more automobiles?

**Tip**: When calculating expected values, make sure each value of the random variable is multiplied by its corresponding probability!

Below is a probability model for the number of automobiles owned by a randomly chosen family in a large town. What is the mean number of automobiles owned?

This problem involves calculating the mean number of automobiles owned by a randomly chosen family using a probability model. The key formula used is the expected value for a discrete probability distribution: μ = Σ (x_i * P(x_i)). The solution demonstrates how to multiply each value by its respective probability and sum the results to obtain a mean of 1.6.

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