For each question on a multiple-choice test, there are four possible answers, of which exactly one is correct. If a student selects answers at random, give the probability that the first question answered correctly is question 3.

Customers arrive at a travel agency at a mean rate of 9 per hour. Assuming that the number of arrivals per hour has a Poisson distribution, give the probability that at least two customers arrive in a given hour.

When a customer buys a product at a supermarket, there is a concern about the item being underweight. Suppose there are 20 “one-pound” packages of frozen ground turkey on display and three of them are underweight. A consumer group buys five of the 20 packages at random. What is the probability that at least one of the five is underweight?

A pair of dice is rolled and X is the random variable defined as the absolute value of the difference of the numbers of dots facing up on two dice. What is the expected value of X? What is the variance of X?

77% of youth now have insurance due to the health care law. Let X equal the number of American youth in a random sample of n = 18 with private health insurance. What is the mean of X?  What is the variance of X?

Exercise 1 Mateo is the star player of a certain soccer team and is getting ready for a very important match after spending several months recovering from an injury. If the player is injured again during this match, the probability that his team wins is 0.32. If Mateo is not injured, the probability that his team loses is 0.18. In addition, according to the team doctor, the probability that the player gets injured during the match is 0.15. a. Draw a probability tree that properly represents the situation described above. Clearly label the probabilities on each branch of the tree. b. What is the probability that Mateo's team wins the match? c. If Mateo's team loses the match, what is the probability that Mateo was injured? d. Consider the event "Mateo is injured" and the event "Mateo's team wins the match." Are these events independent? Clearly justify your answer.

A company devoted to the production and distribution of craft beer has decided to run a quality-control check on a batch of 330 mL Porter beer bottles. A random sample of the contents of 52 bottles was taken; the volume (in millilitres, mL) was measured and is shown below: Dato Volumen (ml) Volumen Volumen Volumen Dato Dato Dato (ml) (ml) (ml) 333 14 326 27 330 40 329 2 328 15 331 28 329 41 327 3 335 16 331 29 329 42 323 4 330 17 333 30 332 43 330 5 331 18 332 31 334 44 332 6 326 19 327 32 336 45 333 7 330 20 330 33 328 46 328 8 329 21 328 34 326 47 329 9 332 22 325 35 330 48 327 10 334 23 330 36 332 49 331 11 333 24 334 37 333 50 333 12 336 25 332 38 331 51 328 13 327 26 325 39 334 52 336

Exercise 4 A company that manufactures engine parts has developed a new type of piston made from aluminium- silicon alloys and other materials. The firm wishes to examine the mechanical properties of these pistons. Specifically, it is interested in evaluating the yield strength of a piston while it is subjected to a constant temperature of 250 °C. Here, the yield strength is the maximum stress, measured in megapascals (MPa), that the piston can withstand before it deforms permanently. From the first production batches, 15 pistons were randomly selected. Each was tested to determine its yield strength (in MPa), giving the following results: 83.2, 90.1, 86.7, 102.4, 95.9, 91.3, 88.1, 84.6, 93.2, 92.6, 100.4, 86.5, 89.2, 96.8, 91.7. Assuming that the yield strength of this type of piston follows a Normal distribution, construct a 96 % confidence interval for the variance, σ², of the pistons' yield strength.

One of the major concerns companies face is achieving the greatest possible standardization of their product characteristics. There are various reasons for seeking to ensure that all product specifications vary as little as possible, such as production costs, industrialization, automation, and others. When large variations appear in products, they can have negative consequences for the company. This was the case for Apple in 2018 with the launch of its new iPad Pro. After its release, multiple buyers reported that, upon taking the new iPad Pro out of its packaging, the device showed a slight curvature. Consequently, that year there was a slight drop in sales of this model. Therefore, Apple has begun publishing the average dimensions of its devices and the standard deviation of these measurements. The average dimensions of the latest iPad Pro are shown in Table 1. iPad Pro dimensions Width Length Thickness Measurement (mm) 178.5 247.6 5.9