A bank accepts rolls of pennies and gives 50 cents credit to a customer without counting the contents. Assume that a roll contains 49 pennies 30 percent of the time, 50 pennies 60 percent of the time, and 51 pennies 10 percent of the time. Estimate the probability that the bank will lose more than 25 cents in 100 rolls.
The solution (first problem part b) has:
$P(Z > 25) = 1 − P(Z ≤ 25) = 1 − N_{20,36}(25) = 1 − N_{0,1}(5/6) = 0.2023$
With $Z$ having a mean of $100\mu = 20$ and variance of $100\sigma^2 = 36$. $\mu$ is the expected loss per roll and $\sigma^2$ the variance of the losses per roll (loss is positive).
However, since $P(Z>25) = P(Z \geq 26) = P(Z^* \geq 1)$, this is equal to $0.1587$. Are both methods valid, as there's a $0.0437$ difference (which to me is quite large) between the two results. Is this an approximation error or am I doing something wrong / unconventional?