Probability：練習問題

Two independent events \(A\) and \(B\) are such that \(P(A) = 0.4\) and \(P(B) = 0.35\).
Find the value of \(P(A \cup B)\).

Two events \(A\) and \(B\) are defined such that \(P(A) = 0.4\), \(P(B) = 0.5\), and \(P(A \cup B) = 0.7\). Find the value of \(P(A' \cap B)\).

In a group of 100 students, 60 study Mathematics (\(M\)), 45 study Physics (\(P\)), and 20 study both subjects. A student is selected at random from the group.
Given that the selected student studies at least one of these two subjects, find the probability that the student studies exactly one of these subjects.

Events \(A\) and \(B\) are mutually exclusive. Given that \(P(A) = 0.5\) and \(P(A \cup B) = 0.8\), find the value of \(P(B')\).

Two events \(A\) and \(B\) are such that \(P(A) = 0.6\), \(P(B'|A) = 0.3\), and \(P(B|A') = 0.4\).
Find the probability \(P(B)\).

The events A and B are mutually exclusive such that \( P(A) = 0.25 \) and \( P(B) = 0.45 \). Calculate \( P(A \cup B) \).

Events S and T are independent. Given that \( P(S) = 0.4 \) and \( P(S \cup T) = 0.7 \), find the value of \( P(T) \).

A box contains 5 gold coins and 3 silver coins. Two coins are taken at random from the box without replacement. Given that at least one of the coins is gold, find the probability that both coins are gold.

In a group of 60 students, 35 study Mathematics, 28 study Physics, and 15 study both subjects. A student is chosen at random from the group. Let \(M\) be the event that the student studies Mathematics and \(P\) be the event that the student studies Physics. Find:
(a) \(P(M \cup P)\)
(b) \(P(M' \cap P)\)
(c) The probability that the student studies neither subject.

A survey was conducted on 120 people regarding their preferred news sources: Television (\(T\)), Internet (\(I\)), and Newspapers (\(N\)). The data is as follows:
- 70 use the Internet.
- 50 use Television.
- 45 use Newspapers.
- 25 use both Internet and Television.
- 20 use both Television and Newspapers.
- 15 use both Internet and Newspapers.
- 10 use all three sources.

(a) Represent this information on a Venn diagram.
(b) Find the probability that a person selected at random uses exactly two news sources.
(c) Given that a person uses the Internet, find the probability that they also use Newspapers.