Riemann surface
matlab

Problem 1.2.1 Solution
(a) An outcome specifies whether the fax is high $(h)$, medium $(m)$, or low $(l)$ speed, and whether the fax has two $(t)$ pages or four $(f)$ pages. The sample space is
$$S={h t, h f, m t, m f, l t, l f} .$$
(b) The event that the fax is medium speed is $A_1={m t, m f}$.
(c) The event that a fax has two pages is $A_2={h t, m t, l t}$.

(a) The sample space of the experiment is
$$S={a a a, a a f, a f a, f a a, f f a, f a f, a f f, f f f} .$$
(b) The event that the circuit from $Z$ fails is
$$Z_F={a a f, a f f, f a f, f f f} .$$
The event that the circuit from $X$ is acceptable is
$$X_A={a a a, a a f, a f a, a f f} .$$
(c) Since $Z_F \cap X_A={$ aaf, aff $} \neq \phi, Z_F$ and $X_A$ are not mutually exclusive.

(d) The event that a fax is either high speed or low speed is $A_3={h t, h f, l t, l f}$.
(e) Since $A_1 \cap A_2={m t}$ and is not empty, $A_1, A_2$, and $A_3$ are not mutually exclusive.
(f) Since
$$A_1 \cup A_2 \cup A_3={h t, h f, m t, m f, l t, l f}=S,$$
the collection $A_1, A_2, A_3$ is collectively exhaustive.

(d) Since $Z_F \cup X_A={a a a$, aaf, af a, aff, faf, fff $} \neq S, Z_F$ and $X_A$ are not collectively exhaustive.
(e) The event that more than one circuit is acceptable is
$$C={a a a, a a f, a f a, f a a} .$$
The event that at least two circuits fail is
$$D={f f a, f a f, a f f, f f f} .$$
(f) Inspection shows that $C \cap D=\phi$ so $C$ and $D$ are mutually exclusive.

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