# Probability a truck full of stones weighs more than 1800kg?

Stones of a particular kind weigh in mean 10 kg and have standard deviation 1 kg. Assume the weight of a truck is normal distributed with mean 1000 kg and standard deviation 100 kg. 50 stones are put into a truck. Determine the probability that the total weight of the 50 stones and the truck is more than 1800 kg. Motivate your calculations.

what I did: $$M = 1000 + (10*50) = 1500$$ $$\sigma = 100 + (1*50) = 150$$ $$P(X > 1800)=1-G(\frac{1800-1500}{150})$$ $$=1 - G(2)$$ $$=1 - (erf(2) + \frac12)$$ $$=\frac12 - (erf(2))$$ $$=\frac12 - .47726$$ $$=.02274$$

Correct answer is $0.0014$. Where did I go wrong?

• You cannot add the $\sigma$'s, I think, but their squares. – Henno Brandsma Nov 26 '14 at 20:27

Let $X$ be the weight of the truck and $Y$ the combined weights of the stones. The variance of $X$ is $10000$ and the variance of $Y$ is $(50)(1^2)$. So the variance of $X+Y$ is $10050$. So the standard deviation of $X+Y$ is $\sqrt{10050}$.
We have assumed that $X$ and $Y$ are independent. We have also assumed that the weights $W_1$ to $W_{50}$ of the $50$ stones are independent.