Fred decides to take a series of n tests, to diagnose whether he has a certain disease

(any individual test is not perfectly reliable, so he hopes to reduce his uncertainty by

taking multiple tests). Let D be the event that he has the disease, p = P(D) be the

prior probability that he has the disease, and q = 1

(a) There are two crimson jars (labeled C1 and C2) and two mauve jars (labeled M1 and

M2). Each jar contains a mixture of green gummi bears and red gummi bears. Show by

example that it is possible that C1 has a much higher percentage of green gummi bears

than M1, and C2 has a much higher percentage of green gummi bears than M2, yet if

the contents of C1 and C2 are merged into a new jar and likewise for M1 and M2, then

the combination of C1 and C2 has a lower percentage of green gummi bears than the

combination of M1 and M2.

(b) Explain how (a) relates to Simpson's paradox, both intuitively and by explicitly

defining events A, B, C as in the statement of Simpson's paradox.

52. As explained in this chapter, Simpson's paradox says that it is possible to have events

A, B, C such that P(A|B,C)

P(A|B) > P(A|Bc).

(a) Can Simpson's paradox occur if A and B are independent? If so, give a concrete

example (with both numbers and an interpretation); if not, prove that it is impossible.

(b) Can Simpson's paradox occur if A and C are independent? If so, give a concrete

example (with both numbers and an interpretation); if not, prove that it is impossible.

(c) Can Simpson's paradox occur if B and C are independent? If so, give a concrete

example (with both numbers and an interpretation); if not, prove that it is impossible.

on Daminov 4. A die is rolled 3 times; find the probability of getting exactly one five. 6 5 5 [ 3573 5. A telemarketing company gets on average 6 orders per 1000 calls. If a company calls 500 people, find the probability of getting 2 orders (Poisson random variable)3 Recruitment of staff at a production company is related to sales figures per month. The probability of increased sales in a randomly selected month is 23% and the probability of unchanged sales is 54%. a) Calculate the probability of reduced sales in a randomly selected month. b) In case of increased sales, the probability of new employment is 76% and in the case of unchanged sales 17%. Calculate the probability of hiring for a randomly selected month. c) Calculate the likelihood of sales being unchanged at the time of new employment. d) 6p For 16 randomly selected months, the average sales value was 4.2 million DKK with a standard deviation of DKK 1.7 million. The industry average for companies in the same The size is SEK 3.6 million per month. Test at 5% significance level if average the sales value of the production company exceeds the industry average.(CO 3) Ten rugby balls are randomly selected from the production line to see if their shape is correct. Over time, the company has found that 89.4% of all their rugby balls have the correct shape. If exactly 6 of the 10 have the right shape, should the company stop the production line? Yes. as the probability of six having the correct shape is not unusual W/No. as the probability of six having the correct shape is unusual W/Yes, as the probability of six having the correct shape is unusual No. as the probability of six having the correct shape is not unusual20. LabTech is a company that manufactures microscopes and other laboratory instruments. On (I point) occasion, one of the microscopes is defective when it comes off of the production line. The probability that a microscope is defective is 0.17%. Find the probability that the first defective microscope comes after the first 100 microscopes off of the production line. 00.156 30.181 00.742 00.844