Variance of linear combination of Normal distributions

Question

A company that develops software received an order for a service to be performed within a week and, in order to decide on the profile of the team of programmers to be used, it should take into account that:

The total number of lines of code (commands, instructions) to be developed is approximately 30 thousand.

The productivity of your most experienced programmers, in hourly commands, follows a with an average of 50 and a standard deviation of 15.

The productivity of its less experienced programmers, in hourly commands, follows a with a mean of 30 and a standard deviation of 10.

Each programmer works six hours a day, five days a week.

What is the probability that the service will be ready in a week if the team consists of 10 more experienced programmers and 20 less experienced programmers?

Solution:

Let P be the team productivity, X the productivity of most experienced programmers and Y the productivity of less experienced programmers:

$X \sim N(50, 15^2)$
$Y \sim N(30, 10^2)$
$P = 10X + 20Y$

Since P is a linear combination of Normal distributions:

$E[P] = 10E[X] + 20E[Y] = 10(50) + 20(30) = 1100$
$Var(P) = 10^2Var[X] + 20^2Var[Y] = 10^2(15^2) + 20^2(10^2) = 62500$

(...)

Question:

The textbook answer says that $Var(P) = 10Var[X] + 20Var[Y] = 10(15^2) + 20(10^2) = 4250$.

Why?

Answer 1

Model 1. Consider two experienced programmers, each producing $X_1 \sim \mathsf{Norm}(\mu = 50, \sigma=15)$ and $X_2 \sim \mathsf{Norm}(\mu = 50, \sigma=15)$ lines of code in an hour, respectively. Then make the assumption that together they produce $X_1 + X_2$ lines of code in and hour. Then $$X_1 + X_2 \sim \mathsf{Norm}\left(\mu=50+50=100,\\ \sigma = \sqrt{15^2+15^2} = 21.2132\right)$$ lines of code are produced in an hour (and we suppose a magic elf instantly puts the lines of code into a coherent whole).

Model 2. By contrast, suppose that, grossly over-caffeinated and working a double rate, the first exerienced programmer can produce $$2X_1 \sim \mathsf{Norm}\left(\mu=2(50)=100,\, \sigma = \sqrt{4*15^2} = 30\right)$$ coherent lines of code in an hour.

Notice that two different probability models have been used here. Hence, I don't believe your expression $P=10X+20Y$ is what the problem means (assuming, contrary to @whubeer's link, that it makes any practical sense at all). I fear for the health of a single programmer (on whatever drugs) working at 20 times normal rate.