An urn contains two blue balls (denoted B, and B,) and three white balls (denoted W,, W2, and W3). One ba recorded, and is replaced. Another ball is then drawn and its color recorded. Let B, W, denote the outcome that the first ball drawn is B, and the second ball drawn is W,. Because the fir ball is drawn, the outcomes of the experiment are equally likely. List all 25 possible outcomes of the experime (a) Consider the event that the first ball that is drawn is blue. Count all the outcomes in this event. What is the total? What is the probability of the event?
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- 1. Suppose that, in Example 2.27, 400 units of food A, 600 units of B, and 600 units of C are placed in the test tube each day and the data on daily food consumption by the bacteria (in units per day) are as shown in Table 2.6. How many bacteria of each strain can coexist in the test tube and consume all of the food? Table 2.6 Bacteria Strain I Bacteria Strain II Bacteria Strain III Food A 1 2 0 Food B 2 1 1 Food C 1 1 2A murder scene is found with two types of blood - that of the victim and that ofthe murderer. As luck would have it, the unidentified blood has an incredibly rareblood disorder, only found in 1 in every million men. The capital and surroundingareas have a population of 20 million - and the police are sure the murderer isfrom the capital. The police have already started cataloging all citizens' bloodtypes for their new super crime-database. They already have nearly 1 millionmale samples in there - and bingo - one man, Mr XY, is a match. He is promptlymarched off to trial, there is no other evidence, but the jury are told that the oddsare 1 in a million that he is innocent. He is duly convicted. The question is, howlikely is it that he did not commit this crime?1) Assuming you have a data matrix X that has n rows and p variables and you know both µ and Σ. How is (X- µ)‘Σ-1(X- µ) distributed? 2) Assuming that you don’t know the values of µ and Σ. How is the statistical distance distributed as n-p gets large?
- A consumer products testing group is evaluating two competing brands of tires, Brand 1 and Brand 2. Tread wear can vary considerably depending on the type of car, and the group is trying to eliminate this effect by installing the two brands on the same 12 cars, chosen at random. In particular, each car has one tire of each brand on its front wheels, with half of the cars chosen at random to have Brand 1 on the left front wheel, and the rest to have Brand 2 there. After all of the cars are driven over the standard test course for 20,000 miles, the amount of tread wear (in inches) is recorded, as shown in the table below. Car 1 2 3 4 5 6 7 8 9 10 11 12 Brand 1 0.64 0.53 0.32 0.61 0.59 0.64 0.34 0.58 0.53 0.43 0.38 0.34 Brand 2 0.47 0.48 0.31 0.37 0.56 0.37 0.30 0.57 0.49 0.41 0.50 0.17 Difference(Brand 1 - Brand 2) 0.17…A purchaser of transistors buys them in lots of 20. It is his policy to randomly inspect 4components from a lot and to accept the lot if at least 3 are nondefective. Suppose eachlot contains exactly five defective transisters. What proportion of lots are rejected?An automobile manufacturer obtains the microprocessors used to regulate fuel consumption in its automobiles from three microelectronic firms: A, B, and C. The quality-control department of the company has determined that 3% of the microprocessors produced by firm A are defective, 4% of those produced by firm B are defective, and 1.5% of those produced by firm C are defective. Firms A, B, and C supply 35%, 20%, and 45%, respectively, of the microprocessors used by the company. What is the probability that a randomly selected automobile manufactured by the company will have a defective microprocessor?
- 1- The number of items produced in a factory during a week is known to be a randomvariable with mean 50● Using Markov's inequality, what can you say about the probability that this week'sproduction exceeds 75?● If the variance of one week's production is equal to 25, then using Chebyshev'sinequality, what can be said about the probability that this week's production isbetween 40 and 60?A manufacturer of DVD players purchases a particular microchip, called LS-24, from three suppliers: Hall Electronics, Schuller Sales, and Crawford Components. 30% of the LS-24 chips were purchased from Hall Electronics, 20% from Schuller Sales, and the remaining 50% from Crawford Components. The manufacturer has extensive track records on all three suppliers and knows that 3% of Hall Electronics' LS-24 chips are defective, 5% of Schuller Sales' LS-24 chips, and 4% of Crawford Components' LS-24 chips are defective. defects. When LS-24 chips arrive at the manufacturer, they are placed directly in a warehouse and are not inspected or identified with the supplier's name. A worker selects a chip to install in a DVD player and finds it defective. What is the probability that it was made by Hall Electronics? What is the probability that Crawford Components made it? What is the probability that it was made by Schuller Sales?The figure to the right shows the results of a survey in which 1012 adults from Country A, 1009 adults from Country B, 1016 adults from Country C, 1010 adults from Country D, and 1005 adults from Country E were asked whether national identity is strongly tied to birthplace. A table labeled "National Identity and Birthplace, People from different countries who believe national identity is strongly tied to birthplace" consists of five rows containing the following information from top to bottom, with row listed first and information listed second: Country A, 31 percent; Country B, 20 percent; Country C, 25 percent; Country D, 53 percent; Country E, 12 percent.Country A31%20%25%53%Country BCountry CCountry DCountry E12% Construct a 99% confidence interval for the population proportion of adults who say national identity is strongly tied to birthplace for each country listed.
- SO what would be the L, Lq, and Wq of this problem? Assuming we are trying to develop and sovle a waiting line system that can accomodate this increased leel of passenger traffic.A k out of n system is one in which there is a group of n components, and the system will function if at least k of the components function. Assume the components function independently of one another. a) In a 3 out of 5 system, each component has probability 0.9 of functioning. What is the probability that the system will function? b) In a 3 out of n system, in which each component has probability 0.9 of functioning, what is the smallest value of n needed so that the probability that the system functions is at least 0.90?Suppose two companies, Company 1 and Company 2, are manufacturing a vaccineto treat COVID-19. Each company ships out 100 vaccines per shipment. A hospital is receiving40% of their vaccine shipments from Company 1 and the remainder of its vaccine shipmentsfrom Company 2. It is reported that from Company 1’s shipments, 7% of the vaccines getsdamaged in shipping meanwhile from Company 2’s shipments, 3% of the vaccines gets damagedin shipping. Suppose the hospital recently received a shipment of 100 vaccines but we are notsure if this shipment came from Company 1 or Company 2. Assuming that the vaccines areindependent of each other, what is the probability that exactly six vaccines in the shipment of100 are damaged?