Solution

Let be the actual amount and the whole dollar amount of check . Then has a uniform distribution on the values 0.00, 0.01, 0.02, …, 0.99. This means that has a uniform distribution on the values -0.50, -0.49,…,-0.01, 0.00, 0.01, …, 0.49. The actual checkbook total is , and Guido’s estimate of the total is. Therefore, the error from Guido’s estimate is .

A easy computation shows that the mean of is -0.005 and its variance is 0.083325.

The probability that the error exceeds 6 is

In the last line the Central Limit Theorem is used to approximate the standardized sum of the by the random variable .