Uncle Henry has been having trouble keeping his weight constant. In fact, during each week, his weight changes from the beginning of the week to the end of the week by a random amount, uniformly distributed between -0.5 and 0.5 pounds. Assuming that his weight change during any given week is independent of his weight change during any other week, approximate the probability that at the end of 50 weeks Uncle Henry will have had a net change in weight of at least +3 pounds. You may want to refer to the standard normal table.

To solve this problem, let's first assume that the weekly weight change for Uncle Henry follows a uniform distribution between -0.5 and 0.5 pounds. Since the weight changes are independent from week to week, we can treat the net change in weight over 50 weeks as the sum of 50 independent random variables, each representing the weight change for one week.

The sum of independent uniform random variables can be approximated by a normal distribution using the Central Limit Theorem.

In this case, the mean of each weekly weight change is 0 pounds (since it's uniformly distributed between -0.5 and 0.5), and the variance of each weekly weight change is given by:
Variance = (0.5 - (-0.5))^2 / 12 = 1/12

Since the net change in weight over 50 weeks is the sum of 50 independent random variables with the same mean and variance, the mean of the net change is 50 * 0 = 0 pounds, and the variance of the net change is 50 * (1/12) = 25/12.

We want to calculate the probability that the net change in weight is at least +3 pounds. To do this, we will standardize the distribution using z-scores and then look up the probability in a standard normal distribution table.

To calculate the z-score, we subtract the mean from the desired value and divide by the standard deviation:
z = (3 - 0) / √(25/12) ≈ 1.73205

Now, we can look up the probability associated with this z-score in the standard normal table. The closest value in the table is 1.73, which corresponds to a probability of approximately 0.9571.

Therefore, the approximate probability that at the end of 50 weeks Uncle Henry will have had a net change in weight of at least +3 pounds is approximately 0.9571.

To solve this problem, we need to calculate the probability that at the end of 50 weeks, Uncle Henry will have a net change in weight of at least +3 pounds.

Step 1: Calculate the standard deviation of weight change during a single week.
Given that the weight change during each week is uniformly distributed between -0.5 and 0.5 pounds, the standard deviation can be calculated as follows:
Standard Deviation (σ) = (b - a) / √12
where a = -0.5 and b = 0.5

Substituting the values, we get:
σ = (0.5 - (-0.5)) / √12
= 1 / √12
= 0.2887 (approx.)

Step 2: Calculate the mean weight change over 50 weeks.
Since the mean weight change during a single week is zero (on average, there is no net weight gain or loss), the mean weight change over 50 weeks will also be zero.

Step 3: Calculate the standard deviation of weight change over 50 weeks.
Since each week's weight change is independent of the others, the standard deviation for 50 weeks can be calculated as the product of the standard deviation for a single week and the square root of the number of weeks:
σ (50 weeks) = √50 * σ (1 week)
= √50 * 0.2887
= 2.0431 (approx.)

Step 4: Calculate the z-score for a net weight change of at least +3 pounds.
The z-score formula is given by:
z = (x - μ) / σ
where x is the net weight change, μ is the mean weight change, and σ is the standard deviation of weight change over 50 weeks.

Substituting the values, we get:
z = (3 - 0) / 2.0431
= 1.4681 (approx.)

Step 5: Use the standard normal table or calculator to find the probability.
Using the standard normal table or a calculator, we can find the probability associated with the z-score of 1.4681. Looking up this value, we find that the probability is approximately 0.9281.

Therefore, the approximate probability that Uncle Henry will have had a net change in weight of at least +3 pounds at the end of 50 weeks is 0.9281.