MAT 4830 Homework 08 Section 4.2, 4.3 Name:_______________________
1. Consider the 20-base sequence
AGGGATACATGACCCATACA. a. Use the first five bases to estimate the four probabilities P ,
A
P ( i

.
P
G
, P
C
, and P . Here,
T b. Repeat part (a) using the first 10 bases. c. Repeat part (a) using all the bases. d. Is there a pattern to the way the probabilities you computed in parts (a-c) changed? If so, what features of the original sequence does this pattern reflect?
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2. Consider the 20-base sequence
CGGTTCGCCTGCGTAGTGCG a. Give the best estimates you can for the probability that each base would appear at site
21. b. Give the best estimates you can for the probabilities of a purine and of a pyrimidine at site 21. c. Which base is most likely to appear at site 21? Is it a purine or a pyrimidine? Does this make sense in light of your answer to part (b)? Explain.
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3. If two events are mutually exclusive, can they also be independent? Explain your mathematical arguments carefully.
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4. (See the definition of independent events on p.132). Show that if events E and F are independent, then the complementary events E' and F' must also be independent.
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5. Medical tests, such as those for diseases, are sometime characterized by their sensitivity and specificity. The sensitivity of a test is the probability that a diseased person will show a positive test result (a correct positive). The specificity of a test is the probability that a healthy person will show a negative test result (a correct negative). a. Both sensitivity and specificity are conditional probabilities. Which of the following are they:
P(- result | disease), P(- result | no disease),
P(+ result | disease), P(+ result | no disease). b. The other conditional probabilities listed in (a) can be interpreted as probabilities of false positives and false negatives. Which is which? c. A study investigated the use of X-ray readings to diagnose tuberculosis. Diagnosis of
1,820 individuals produced the data in the following table . Compute both the sensitivity and specificity for this method of diagnosis.
Persons without TB Persons with TB
Negative X-ray 1,739 8
Positive X-ray 51 22
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6. Ideally, the specificity and sensitivity of medical tests should be high (close to 1).
However, even with a highly specific and sensitive test, screening a large population for a disease that is rare can produce surprising results. a. Suppose the sensitivity and specificity of a test for disease are both 0.99
. The test is applied to everyone in a population of 100,000 individuals, only 100 of whom have the disease. Compute how many individuals with/without the disease you would expect to test positive/negative. Organize your results in a table like that in the preceding problem. b. Use the table you produced in part (a) to compute the conditional probability that a person who tests positive actually has the disease. Comment on the results.
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