Probability：练习题

A fair coin is tossed and a fair six-sided die is rolled. Find the probability of obtaining a head on the coin and a prime number on the die.

Three different mathematics books and two different biology books are placed in a row on a shelf. Find the number of ways to arrange the books such that the three mathematics books are placed together.

A group of 9 people consists of 5 men and 4 women. A committee of 4 people is to be selected from this group. Find the number of ways the committee can be formed if it must contain at least one man and at least one woman.

For two events \( A \) and \( B \), it is given that \( P(A) = 0.4 \), \( P(B) = 0.5 \), and \( P(A \cap B) = 0.2 \). Find the value of \( P(A \cup B) \).

For two independent events \( A \) and \( B \), it is given that \( P(A) = 0.3 \) and \( P(A \cup B) = 0.58 \). Find the value of \( P(B) \).

Given that the probability of event \( A \) occurring is \( 0.35 \), find the probability that event \( A \) does not occur.

In a class, \( 60\% \) of students like Math, \( 40\% \) like Science, and \( 20\% \) like both. Find the probability that a student likes Science given that they like Math.

Find the number of ways to arrange the letters in the word CHAIR such that the vowels A and I are always separated.

A group of 7 students, consisting of 4 boys and 3 girls, are to be seated in a row for a photograph. Find the number of ways the students can be arranged if the 3 girls must always sit together.

In a class of 30 students, 18 play football, 15 play basketball, and 8 play both sports. A student is chosen at random from the class.
(a) Draw a Venn diagram to represent this information.
(b) Find the probability that the student plays only football.
(c) Given that the chosen student plays basketball, find the probability that he also plays football.