Exercise 1: There are 40 multiple choice questions in a history test. For each question, there are 5 alterna- tives of which only one alternative is the correct answer. 3 marks are awarded for each correct answer, and 1 mark is deducted for each wrong answer. Little John is taking this history test. However, he is not fully prepared. For each question, little John either knows the answer or he doesn't. If he doesn't, then he will guess the answer at random. Suppose that he knows the answer to each question with probability 0.55. It is assumed that he answers each question independently and attempts all the questions.
Let the random variable X denote the number of questions he answers correctly. Then X is a
Binomial distribution with parameters n = and p = .
(Round your answer to two decimal places. For example: 3.00; 3.12; 0.10)
Exercise 2: Little John's expected score in the test mentioned in Exercise 1 is .
(Round your answer to two decimal places. For example: 3.00; 3.12; 0.10)
Exercise 3: A life insurance company estimates that the probability that an individual in a certain risk group will survive one year is 0:99. Such a person wishes to buy a $75,000 one-year term life insurance policy. This means that if the person dies in the next year the company will pay $75,000. Let C denote how much the insurance company charges for such a policy. If X denotes the net pro t to the company from selling the policy, what is the probability function of X?
Exercise 4: For Exercise 3, the value that C should be for the expected net pro t to be zero is .
(Round your answer to two decimal places. For example: 3.00; 3.12; 0.10)
Exercise 5: Suppose 10 people buy such a policy as mentioned in Exercise 3. Using the law of rare events or otherwise, what is the approximate probability that exactly 2 of them die in the next year? Pick the option closest to the answer.
(a) 0.1
(b) 0.3
(c) 0.5
(d) 0.7
(e) 0.9
Exercise 6: In 30 independent ips of a fair coin, the expected number of runs in the sequence is .
(Round your answer to two decimal places. For example: 3.00; 3.12; 0.10)
Exercise 7: Investigators need to determine which of 500 adults have a certain medical condition that a ects 2% of the population. A blood sample is taken from each of the individuals. Assume that the 500 adults are a random sample from the population at large. Instead of testing all 500 blood samples, investigators group the samples into 50 groups of 10 each, mix a little of the blood from each of the 10 samples in each group, and test each of the 50 mixtures. What is the probability that any such mixture will contain the blood of at least one diseased person and hence test positive?
Pick the option closest to the answer.
(a) 0.1
(b) 0.3
(c) 0.5
(d) 0.7
(e) 0.9
Exercise 8: In the 50 mixtures of blood as mentioned in Exercise 7, the expected number of mixtures that will test positive is . (Round your answer to two decimal places. For example: 3.00; 3.12; 0.10)