Page 85 - Applied statistics and probability for engineers

Section 2-8/Random Variables 63

2-222. A company that tracks the use of its Web site determined (b) If a password consists of exactly 6 letters and 1 number,
that the more pages a visitor views, the more likely the visitor how many passwords are possible?
is to provide contact information. Use the following tables to (c) If a password consists of 5 letters followed by 2 numbers,
answer the questions: how many passwords are possible?
2-226. Natural red hair consists of two genes. People with red
Number of pages viewed: 1 2 3 4 or more hair have two dominant genes, two regressive genes, or one
Percentage of visitors: 40 30 20 10 dominant and one regressive gene. A group of 1000 people was
categorized as follows:
Percentage of
visitors in each Gene 2
page-view category
that provides Gene 1 Dominant Regressive Other
contact information: 10 10 20 40 Dominant 5 25 30
Regressive 7 63 35
(a) What is the probability that a visitor to the Web site pro-
vides contact information? Other 20 15 800
(b) If a visitor provides contact information, what is the prob- Let A denote the event that a person has a dominant red hair
ability that the visitor viewed four or more pages? gene, and let B denote the event that a person has a regressive
2-223. An article in Genome Research [“An Assessment of red hair gene. If a person is selected at random from this group,
Gene Prediction Accuracy in Large DNA Sequences” (2000, compute the following: ( B) ( B)
Vol. 10, pp. 1631–1642)], considered the accuracy of com- (a) P A ( ) (b) P A∩ (c) P A∪
(
| (
mercial software to predict nucleotides in gene sequences. The (d) P A′ ′ B) (e) P A B)
following table shows the number of sequences for which the (f) Probability that the selected person has red hair
programs produced predictions and the number of nucleotides
correctly predicted (computed globally from the total number 2-227. Two suppliers each supplied 2000 parts that were
of prediction successes and failures on all sequences). evaluated for conformance to speciications. One part type
was more complex than the other. The proportion of noncon-
Number of forming parts of each type are shown in the table.
Sequences Proportion
Simple Complex
GenScan 177 0.93 Supplier Component Assembly Total
Blastx default 175 0.91 1 Nonconforming 2 10 12
Blastx topcomboN 174 0.97 Total 1000 1000 2000
Blastx 2 stages 175 0.90
2 Nonconforming 4 6 10
GeneWise 177 0.98
Total 1600 400 2000
Procrustes 177 0.93 One part is selected at random from each supplier. For each
Assume the prediction successes and failures are independent supplier, separately calculate the following probabilities:
among the programs. (a) What is the probability a part conforms to speciications?
(a) What is the probability that all programs predict a nucleotide (b) What is the probability a part conforms to speciications
correctly? given it is a complex assembly?
(b) What is the probability that all programs predict a nucleotide (c) What is the probability a part conforms to speciications
incorrectly? given it is a simple component?
(c) What is the probability that at least one Blastx program (d) Compare your answers for each supplier in part (a) to those in
predicts a nucleotide correctly? parts (b) and (c) and explain any unusual results.
2-224. A batch contains 36 bacteria cells. Assume that 12 2-228. Consider the treatments in Exercise 2-57. Suppose a
of the cells are not capable of cellular replication. Of the cells, patient is selected randomly. Let A denote the event that the
6 are selected at random, without replacement, to be checked patient is treated with ribavirin plus interferon alfa or interferon
for replication. alfa, and let B denote the event that the response is complete.
(a) What is the probability that all 6 of the selected cells are Determine the following probabilities.
able to replicate? (a) P(A | B) (b) P(B | A) (c) P A( ∩ B) (d) P A( ∪ B)
(b) What is the probability that at least 1 of the selected cells is 2-229. Consider the patient groups in Exercise 2-98. Suppose
not capable of replication? a patient is selected randomly. Let A denote the event that the
2-225. A computer system uses passwords that are exactly patient is in group 1 or 2, and let B denote the event that there is
seven characters, and each character is one of the 26 letters no progression. Determine the following probabilities:
(a–z) or 10 integers (0–9). Uppercase letters are not used.
(a) How many passwords are possible? (a) P(A | B) (b) P(B | A) (c) P A( ∩ B) (d) P A( ∪ B)