Problem Sets: Events, Probability, Counting Methods (随机事件、概率、计数方法)

The probability \(p\) of event \(A\) that at least two people in a group of \(k\)  people will have the same birthday (the same day of the same month but not necessarily in the same year).

There are four students in a dormitory. What is the probability that exactly three students have their birthdays in the same month? What is the probability that everyone’s birthday in different months?

There are 5 different books. Suppose we want to put them on a bookshelf. What is the possibility that the order of the five books is exactly 1, 2, 3, 4, 5 from left to right?

Suppose there are 30 black cell phones and 40 white cell phones in a store. An employee takes 15 phones at random. What is the probability that the employee holds 8 black phones exactly among the chosen phones?

Suppose that \(\operatorname{Pr}(A\cap B) = 0.3\), \(\operatorname{Pr}(B) = 0.5\), and \(\operatorname{Pr}(A\cup B) = 0.65\), then what is the value of \(\operatorname{Pr}(A\cup B^{c})\)?

Suppose that there are 120 balls of 3 colors in a box. If it is known that there are 30 red balls and 40 blue balls, 30 balls have been selected at random, what is the probability that 10 balls of each color have been selected?

Suppose that there are 20 balls of two colors in a box, among which 4 are red and 16 are blue. Now, all the 20 balls are distributed randomly into 4 boxes, with each box containing exactly 5 balls. What is the probability that each box contains only 1 red ball and 4 blue balls?

There are \(N\) products, and \(M\) of them are defective. If we randomly select \(n\) products, what is the probability that exactly \(m\) of the selected products are defective?

There are \(n\) boys and \(n\) girls. If they are randomly arranged in a queue. What is the probability that no two girls are adjacent? What if they are randomly arranged in a circle?

\(n\) Books are randomly distributed between two people. What is the probability that each person receives at least one book?

There are two boxes of matches in a pocket, each containing \(n\) matches. Every time, one box is chosen at random, and a match is used. At some point, a box is selected and found to be empty. What is the probability that the other box contains exactly \(m\) matches?

What is the general formula for the number of unordered samples of size \(k\) with replacement from \(n\) elements?

Suppose there are 4 white flags, 4 red flags, and 2 blue flags. If all flags of the same color are identical, how many distinct arrangements can be formed by hanging a line of 10 flags?

Suppose that 13 cards are selected at random from a regular deck of 52 playing cards. If it is known that at least one ace has been selected, what is the probability that at least two aces have been selected?

Let \(A\) and \(B\) be two events. Suppose that the probability that neither event occurs is \(\frac{𝟑}{8}\). What is the probability that at least one of the events occurs?

Let \(A\) and \(B\) be two events. Suppose \(\operatorname{Pr}(A) = 0.5\), \(\operatorname{Pr}(A \cap B) = 0.2\), and \(\operatorname{Pr}(A \cup D) 0.4\). What is \(\operatorname{Pr}(D)\)?

Consider two events \( A \) and \( B \) such that \( \Pr(A) = 0.325 \) and \( \Pr(B) = 0.585 \). Determine the value of \( \Pr(A^c \cap B) \) if \( A \) and \( B \) are disjoint.