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We are given that the weight loss is uniformly distributed between 6 pounds and 12 pounds, meaning that each value within this range has an equal probability of occurring. This makes it a **uniform distribution problem**.

The uniform distribution's probability density function (PDF) for a random variable \(X\) (representing weight loss) that is uniformly distributed between \(a = 6\) and \(b = 12\) is given by:
\[
f(x) = \frac{1}{b - a} = \frac{1}{12 - 6} = \frac{1}{6}
\]
This means the probability density is \( \frac{1}{6} \) for any interval within the range from 6 to 12 pounds.

Now, let’s solve the two parts of the problem:

### 3. Probability of losing **less than 11 pounds**:
We need to find \( P(X < 11) \).

The cumulative probability for a uniform distribution is calculated as:
\[
P(X < x) = \frac{x - a}{b - a}
\]
Here, \(a = 6\), \(b = 12\), and \(x = 11\). Substituting the values:
\[
P(X < 11) = \frac{11 - 6}{12 - 6} = \frac{5}{6}
\]
So, the probability of losing less than 11 pounds is:
\[
P(X < 11) = \frac{5}{6} \approx 0.8333
\]
Thus, the probability is **0.8333**, or **83.33%**.

### 4. Probability of losing **between 8.5 pounds and 10 pounds**:
We need to find \( P(8.5 < X < 10) \).

For this, we calculate the probability by finding the cumulative probabilities at 8.5 and 10 pounds, and then subtracting:
\[
P(8.5 < X < 10) = P(X < 10) - P(X < 8.5)
\]
Using the cumulative probability formula again:
\[
P(X < 10) = \frac{10 - 6}{12 - 6} = \frac{4}{6} = \frac{2}{3} \approx 0.6667
\]
\[
P(X < 8.5) = \frac{8.5 - 6}{12 - 6} = \frac{2.5}{6} \approx 0.4167
\]
Now subtract:
\[
P(8.5 < X < 10) = 0.6667 - 0.4167 = 0.25
\]
Thus, the probability of losing between 8.5 and 10 pounds is **0.25**, or **25%**.

### Summary of Results:
1. Probability of losing less than 11 pounds = 0.8333 (83.33%).
2. Probability of losing between 8.5 and 10 pounds = 0.25 (25%).

Let me know if you need more details or have any questions. Here are 5 related questions that expand on this problem:

1. How would the results change if the weight loss range was 5 to 15 pounds instead of 6 to 12?
2. How do you calculate the mean of a uniform distribution?
3. What is the probability of losing exactly 9 pounds in a uniform distribution?
4. How would you calculate the standard deviation for this uniform distribution?
5. If a person lost more than 11 pounds, what would be the probability of that happening?

**Tip**: In a uniform distribution, the probability of any single exact value (e.g., exactly 9 pounds) is always 0, because the area under a point in a continuous distribution is zero.

Assume that the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread evenly over the range of possibilities, so that there is a uniform distribution. Find the probability of the given range of pounds lost:

3. Less than 11 pounds
4. Between 8.5 pounds and 10 pounds

Learn how to calculate probabilities in a uniform distribution context, specifically for a weight loss range between 6 and 12 pounds. This step-by-step solution shows how to find the probability of losing less than 11 pounds and between 8.5 and 10 pounds.

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