https://file.aiquickdraw.com/mathbot/file/1727707175092worwtb1w.jpg

Let's break down the tasks and solve the problems step by step based on the provided image.

### Part 1: Mean and Standard Deviation of the Probability Distribution

You are given a probability distribution of \( x \), representing the number of girls in 5 births, with corresponding probabilities \( P(x) \):

| \( x \) | \( P(x) \) |
| ------ | ----------- |
| 0      | 0.031       |
| 1      | 0.156       |
| 2      | 0.313       |
| 3      | 0.313       |
| 4      | 0.156       |
| 5      | 0.031       |

We are tasked with finding the mean (\( \mu \)) and the standard deviation (\( \sigma \)) of this distribution.

#### 1.1: Finding the Mean (\( \mu \))

The formula for the mean of a discrete probability distribution is:

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

#### 1.2: Finding the Standard Deviation (\( \sigma \))

The standard deviation is calculated using:

\[
\sigma = \sqrt{\sum [(x - \mu)^2 \cdot P(x)]}
\]

I'll calculate both the mean and standard deviation now.The calculated mean (\( \mu \)) of the probability distribution is:

\[
\mu = 2.5
\]

The calculated standard deviation (\( \sigma \)) is:

\[
\sigma = 1.1162
\]

### Part 2: Calculate the Required Probabilities

Now let's solve the following probabilities based on the distribution:

- \( \text{P(more than 3 girls)} \)
- \( \text{P(at least 3 girls)} \)
- \( \text{P(an odd number of girls)} \)
- \( \text{P(fewer than 2 girls)} \)
- \( \text{P(at most 1 girl)} \)

I'll calculate these probabilities now using the provided distribution.Here are the calculated probabilities based on the given distribution:

- **P(more than 3 girls):** 0.187
- **P(at least 3 girls):** 0.500
- **P(an odd number of girls):** 0.500
- **P(fewer than 2 girls):** 0.187
- **P(at most 1 girl):** 0.187

Would you like further details or clarifications on any of the calculations?

### Related Questions:
1. How would the mean and standard deviation change if the probability distribution shifted?
2. How is the probability of getting an odd number of girls equal to that of at least 3 girls?
3. What is the probability of getting exactly 3 girls?
4. How can you visualize this probability distribution effectively?
5. How do you interpret the standard deviation in the context of this problem?

### Tip:
The sum of all probabilities in a valid probability distribution should always equal 1.

This worksheet involves calculating the mean, standard deviation, and specific probabilities from a discrete probability distribution. The data represents the number of girls in 5 births. Concepts such as P(more than 3 girls), P(at least 3 girls), and P(fewer than 2 girls) are covered.

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