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I see six potential reasons for this. You've already mentioned one, the impossibility of "uniformly" distributing the satellites over a sphere, in the question, and percusse and Tim mentioned three mo …

Your method of apportionment is the one that John Quincy Adams proposed in 1832. It has never been used to apportion seats in the U.S. House of Representatives. Most methods of apportionment in wides …

The probability that three components are still working after time $t$ is $\mathrm e^{-3\lambda t}$. The probability density for the first of them to fail at time $t$ is the negative derivative of thi …

Life as we know it in three spatial dimensions (four space-time dimensions) is only possible because a number of things coincide at this dimensionality. Planetary orbits are only stable in a space-tim …

I'm not sure whether you'd count this: The fact that an integral number of identical circles fit around a circle in the plane leads to drains being designed like this: Assuming the aim is to have as …

No, this is not the way to go about this. You got the same proportion because you reduced both numbers by the same proportion. You’re not using the data that would be required (at a minimum) to draw t …

There are two differences. One is that the second matrix is $4\times4$ because it represents an affine transform, which can accommodate translations but in this case doesn't; the fourth component does …

There's no justification for the "given that the lifetime of machine $1$ is greater than $t$" part. This is not given. It should be and, as in your first paraphrase. If $P(M_1\gt t)$ is very low, the …

The statements in the book are correct if you interpret "of size" to mean "of size at most". Taken literally, they are clearly false.

You could extend your approach for $\mu=0$ to general $\mu$ in order to reduce the problem from $3$ to $1$ dimensions, and then estimate the remaining degree of freedom e.g. using expectation maximiza …