<s>[INST] Probability - Peculiar die 
A peculiar die has the following properties: on any roll the probability of rolling either a $4$, a $6$, or a $1$ is $1/2$, just as it is with an ordinary die. Moreover, the probability of rolling either a $1$, a $3$, or a $2$ is again $1/2$. However, the probability of rolling a $1$ is $5/16$, not $1/6$ as one would expect of an ordinary fair die.
From what you know about this peculiar die, compute the following:
a.    The probability of rolling either a $5$, a $3$, or a $2$;
b.    The probability of rolling a $5$.

My answer
Given $P(1) = \frac{5}{16}$, then
$$P(2) + P(3) + P(1) = P(2) + P(3) + \frac{5}{16} = \frac{1}{2} \iff P(2) + P(3) = \frac{3}{16}$$
Similarly, using $P(4) + P(6) + P(1) = \frac{1}{2}$, I find
$$P(6) + P(4) = \frac{3}{16}$$
b. $P(5) = 1 - \left(\frac{5}{16} + \frac{3}{16} + \frac{3}{16}\right) = \frac{5}{16}$
a. $P(5) + P(3) + P(2) = \frac{5}{16} + \frac{3}{16} = \frac{1}{2}$
Are my calculations correct?
 [/INST] Your calculations are correct but slightly more complicated than necessary.
In a way, the order in which parts (a) and (b) are presented is a hint at a simpler solution. 
Letting $R_{i,j,\ldots,k}$ be the event that we roll one of the numbers in the set $\{i,j,\ldots,k\},$
then we are given that $R_{1,4,6} = R_{1,2,3} = \frac12$ and that
$R_1 = \frac{5}{16}.$
For part (a), simply observe that $\{2,3,5\}$ and $\{1,4,6\}$ are disjoint and that $\{2,3,5\} \cup \{1,4,6\}$ is the entire sample space.
Therefore
$$P(R_{2,3,5}) = 1 - P(R_{1,4,6}) = 1 - \tfrac12 = \tfrac12.$$
Now for part (b) you know that 
$$P(R_1) + P(R_{2,3}) = P(R_{1,2,3}) = \tfrac12$$
and 
$$P(R_{2,3}) + P(R_5) = P(R_{2,3,5}) = \tfrac12,$$ 
so
$$P(R_{2,3}) + P(R_5) = P(R_1) + P(R_{2,3}),$$
and canceling $P(R_{2,3})$ on each side you have
$$P(R_5) = P(R_1) = \tfrac{5}{16}.$$
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