1. Suppose in one box there are 100 pens and 6 of them are damaged. In the process of randomly selecting 5 pens, find the probabilities a) All 5 damaged pens were selected. b) 3 good pens and 2 damaged pens will be selected. 2. A study was conducted to obtain students' views on the implementation of virtual teaching in three colleges of study X, Y and Z. Students will choose whether they agree, disagree or neutral regarding the implementation of virtual teaching. TABLE 1 provides information related to the study. TABLE 1 Student Answers Agree Do not agree Neutral College X 815 2000 135 College Y 455 1500 545 College Z 530 1300 720 Suppose a student is randomly selected. a) Find the probability of students agreeing with the implementation of virtual teaching. b) Find the probability of students agreeing from among the students of College Z 3. A club provides three games namely badminton (B), tennis (T) and football (BS). Membership for badminton, tennis and football each is a total of 30 people and is depicted in the Venn diagram as in FIGURE1. FIGURE 1 18 Q BS a) Find the values of P, Q and R. b) If a badminton player is randomly selected, find the value of the probability that he plays tennis but does not play football. 4. The discrete random variable Y has a probability distribution as in TABLE 2. 2 3 4 P( Y=V) 0.2 0.43 0.25 0.1 a) Sketch a graph of the probability distribution function. b) Find P (X = 1). c) Calculate the expected value of Y, E (Y). d) Calculate the value of the variance for Y, Var (Y).5. A box contains three red marbles and two yellow marbles. Two balls are removed one after the other without replacement from the box. Assume that the discrete variable X represents the number of red marbles obtained. Prove that X is a discrete random variable and form a table of probability distributions. 6. Ten percent of all air conditioners manufactured by a factory are imperfect and unsuitable for market. Five air conditioners were randomly selected and inspected to identify the quality of the equipment. a) What is the value of the probability that an air conditioner is imperfect? b) What is the value of the probability that at least three air conditioners are imperfect? 7. A water filter machine salesman in CHUWAR company on average sells 7 water filter machines in a week. a) Find the value of the probability that there are 3 water filter machines can be sold in a week. b) Find the value of the probability that there is only one water filter machine that can be sold in a day. c) Calculate the average value and standard deviation of car sales in a month. The average marks for students taking the mid -semester examination for the Introductory Statistics course are assumed to be normally distributed with the average mark being 60 and the standard deviation being 10. a) Find the percentage of students who get examination marks more than 70. b) Find the percentage of students who get examination marks between 50 and 90. 9. A study conducted found that sixty percent of residents in a housing estate use food delivery services at least once a month. A random sample of 100 residents was selected from among those who use this food delivery service. Calculate the average and standard deviation of the sampling distribution for this sample and comment on the shape of the sampling distribution