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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*

*T*

. Here, 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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