https://file.aiquickdraw.com/mathbot/file/1726784401527lq61x31k.jpg

It seems the file you uploaded is an image (`.jpg` format), not an Excel file. However, I can still help you based on the data visible in the image.

### Data from the Image:
The given probability distribution table is as follows:

| \( x \) (New Clients) | \( f(x) \) (Probability) |
|-----------------------|-------------------------|
| 20                    | 0.05                    |
| 25                    | 0.20                    |
| 30                    | 0.25                    |
| 35                    | 0.15                    |
| 40                    | 0.15                    |
| 45                    | 0.10                    |
| 50                    | 0.10                    |

### Question (a): Is this a valid probability distribution?

To check if it's a valid probability distribution:
1. Each probability \( f(x) \) must be between 0 and 1.
2. The sum of all probabilities must equal 1.

### Question (b): Probability of obtaining 40 or more new clients

To calculate this, sum the probabilities for \( x = 40 \), \( x = 45 \), and \( x = 50 \).

I'll now perform the necessary calculations.### Answer (a): Is this a valid probability distribution?
Yes, this is a valid probability distribution because:
- Each probability \( f(x) \) is between 0 and 1.
- The sum of all probabilities is exactly 1.

### Answer (b): What is the probability that Backen and Hayes LLC will obtain 40 or more new clients?
The probability that they will obtain 40 or more new clients is **0.35** (or 35%).

Would you like further clarification or details on any part? Here are a few related questions to expand on:

1. How can we compute the expected number of new clients using this distribution?
2. What is the variance of the number of new clients in this distribution?
3. How would the probability change if we wanted 30 or more new clients?
4. Could this distribution be used to model demand for a different business service?
5. How would you determine if this distribution is skewed or symmetric?

**Tip**: Always verify that the sum of probabilities is 1 when working with probability distributions to ensure they are valid.

Is this a valid probability distribution? What is the probability that Backen and Hayes LLC will obtain 40 or more new clients?

This problem examines a probability distribution to check its validity and calculates the probability that Backen and Hayes LLC will have 40 or more clients. The data involves summing specific probabilities, suitable for students learning about probability theory.

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