Set functions, sample spaces, events, and axioms of probability：练习题

A finite sample space \(S\) consists of four mutually exclusive outcomes \(\{e_1, e_2, e_3, e_4\}\). The probabilities are assigned such that \(P(e_1) = 0.1\), \(P(e_3) = 0.3\), and \(P(e_2) = 2P(e_4)\). Find the probability of the event \(E = \{e_2, e_4\}\).

Consider three events \(A\), \(B\), and \(C\) with \(P(A) = 0.5\), \(P(B) = 0.6\), and \(P(C) = 0.4\). Suppose \(P(A \cap B) = 0.3\), \(P(A \cap C) = 0.2\), \(P(B \cap C) = 0.24\), and \(P(A \cap B \cap C) = 0.1\). Determine the probability that none of the three events occur.

Let \(A_1, A_2, A_3, \dots\) be an infinite sequence of mutually exclusive events whose union is the entire sample space \(S\). Suppose that for some constant \(k\), the probability of each event is given by \(P(A_n) = k \cdot (1/3)^n\) for \(n = 1, 2, 3, \dots\). Find the probability of the event \(A_1 \cup A_2\).

Suppose that for two events \(A\) and \(B\), we have \(P(A) = 0.5\), \(P(B) = 0.4\), and \(P(A \cap B) = 0.1\). Calculate the probability of the event \(A \cup B^c\).

A biased coin with probability of heads \(p = 0.6\) is tossed repeatedly until a head appears. Let \(k\) be the number of tails observed before the first head. Find the probability that \(k\) is an even number (note that 0 is even).

Let the sample space be the set of all positive integers \(S = \{1, 2, 3, \dots\}\). A probability measure is defined on \(S\) such that the probability of an outcome \(n\) is given by \(P(\{n\}) = \frac{c}{3^n}\) for \(n = 1, 2, 3, \dots\), where \(c\) is a constant. Consider the following two events:
\(A\): the outcome is an even integer.
\(B\): the outcome is a multiple of 3.
Calculate the probability of the event \(A \cup B^c\).