Section A - Solve the following problems-
1. Exercise 1 in page 17
When events are Non-mutually exclusive, two or more of them can happen at the same time. In this case the general rule of addition can be used in calculating probabilities.
General Rule of addition when events are non-mutually exclusive
P(A U B) = P(A) + P(B) - P(A ∩ B)
Exercise 1. Calculate the following probabilities, either by Venn diagram or Probability Rules
a)? P (female)
b)? P (16 or older)
c)? P (female and under 16)
d)? P (A' and B')
e)? P (A' or B')
2. Example 5 and Exercise 1 in page 19
Example 5. Suppose that a large civil engineering company is involved in two major projects: the construction of a bridge in Brazil and a dam in Russia. It is estimated that the probability that the bridge construction will be completed on time is 0.8, while the probability that the dam will be completed on time is 0.7. The teams involved with the two projects operate totally independently, and the company wants to determine the probability that both projects will be completed on time.
Exercise 1. In a Western community, the probability of passing the road test for a driver's license on the first try is 0.75. After that, the probability of passing becomes 0.60 regardless of how many times a person has failed. What is the probability of getting one's license on the fourth try? (Hint: you can use a tree diagram)
3. Exercise 2 in page 21
Exercise 2. In reference to example (e) find P (G|N') from the table and by using the formula of conditional probability.
4. Exercise 1 and 2 in page 26
Exercise 1. The probability is 0.60 that a famous Nigerian distance runner will enter the Boston marathon. If he does not enter, the probability that last year's winner will repeat is 0.66, but if he enters, the probability that last year's winner will repeat is only 0.18. What is the probability that last year's winner will repeat?
Exercise 2. No diagnostic test is infallible, so imagine that the probability is 0.95 that a certain test will correctly diagnose a person with diabetes as being diabetic, and it is 0.05 that the test will incorrectly diagnose a person without diabetes as being diabetic. It is know that roughly 10% of the population is diabetic. Estimate the probability that a person diagnosed as being diabetic actually has diabetes. Use Bayes' theorem.
5. Exercise 3 in page 29
Exercise 3. Two ?rms A and B consider bidding on a road-building job that may or may not be awarded depending on the amount of the bids. Firm A submits a bid and the probability is ¾ that it will get the job provided ?rm B does not bid. The odds are 3 to 1 that B will bid, and if it does, the probability that A will get the job is only ?.
(a) What is the probability that A will get the job?
(b) If A gets the job, what is the probability that B did not bid?
6. Example 2 in page 27:
Example 2. Suppose that in a particular chest clinic 5% of all patients who have been to the clinic are ultimately diagnosed as having lung cancer, while 50% of patient are smokers. By considering the records of all patients previously diagnosed with lung cancer, we know that 80% were smokers. A new patient comes into the clinic. We discover this patient is a smoker. What we want to know is the probability that this patient will be diagnosed as having lung cancer.
7. Exercise 4 in page 29:
Exercise 4. In a cannery, assembly lines, I, II, and III account for 50, 30, and 20% of the total output. If 0.4% of the cans from assembly line I are improperly sealed, and the corresponding percentages for assembly lines II and III are 0.6% and 1.2%, what is the probability that an improperly can (discovered at the ?nal inspec?on of outgoing products) will have come from assembly line I ?
Section B - In reference to Example 6 in page 22 -
A black box contains five identical balls, two red, two blue and one white. Two balls are drawn at random (sequentially, one after the other).
B1. What is the probability that they will both be blue if
a) the first ball is put back before the second ball is drawn
b) the first ball is not put back before the second ball is drawn
B2. What is the probability that they will be the first red and second white if
a) the first ball is put back before the second ball is drawn
b) the first ball is not put back before the second ball is drawn.
Attachment:- Assignment.rar