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.
To answer this question, we can use the concept of random variables and the properties of independent, identically distributed variables.
First, let's define a random variable X representing the weight change during a single week. Based on the given information, X follows a uniform distribution between -0.5 and 0.5 pounds.
Next, let's define another random variable S representing the net weight change after 50 weeks. Since weight changes during different weeks are independent, the total net weight change is the sum of the individual weekly weight changes. Therefore, S is the sum of 50 independent, identically distributed variables X.
Now, we need to find the probability that S is greater than or equal to +3 pounds after 50 weeks. To do this, we can use the Central Limit Theorem, which tells us that the sum of a large number of independent and identically distributed variables will be approximately normally distributed.
The mean of S is 50 times the mean of X, which is 50 * (0 + (-0.5)) / 2 = -12.5 pounds.
The standard deviation of S is the square root of 50 times the variance of X, which is sqrt(50 * (0.5 - (-0.5))^2 / 12) = sqrt(50 * 1/12) ≈ 3.42 pounds.
Next, we can approximate the probability using the standard normal distribution. We convert +3 pounds to a z-score using the formula: z = (x - mean) / standard deviation. In this case, x = 3, mean ≈ -12.5, and standard deviation ≈ 3.42.
Therefore, z = (3 - (-12.5)) / 3.42 ≈ 5.39.
We can then use a z-table or a calculator to find the corresponding probability. Looking up the z-score 5.39 in the table, we find that the probability is approximately 1.
Thus, 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 1.