1. Xuan Guo xguo9@student.gsu.edu
Lab 12
Xuan Guo

2. Question 9 Section 7.3
Suppose that 8% of the patients tested in a clinic are infected with HIV. Furthermore, suppose that when a blood test for HIV is given, 98% of the patients infected with HIV test positive and that 3% of the patients not infected with HIV test positive. What is the probability that
a) a patient testing positive for HIV with this test is infected with it?
b)a patient testing positive for HIV with this test is not
infected with it?
c) a patient testing negative for HIV with this test is infected with it?
d)a patient testing negative for HIV with this test is not

3. Question 9 Section 7.3
Suppose that 8% of the patients tested in a clinic are infected with HIV. Furthermore, suppose that when a blood test for HIV is given, 98% of the patients infected with HIV test positive and that 3% of the patients not infected with HIV test positive. What is the probability that
a) a patient testing positive for HIV with this test is infected with it?
0.7396
b)a patient testing positive for HIV with this test is not
infected with it?
0.2604
c) a patient testing negative for HIV with this test is infected with it?
0.002
d)a patient testing negative for HIV with this test is not infected with it?
0.998

4. Question 11 Section 7.4
Suppose that we roll a fair die until a 6 comes up or we have rolled it 10 times. What is the expected number of times we roll the die?

5. Question 11 Section 7.4
Suppose that we roll a fair die until a 6 comes up or we have rolled it 10 times. What is the expected number of times we roll the die?
P(X = k) = (1-1/6)k-11/6
K = 1 1/6
K = 2 5/6 * 1/6
K = 3 …
K = 4 …
K = 5 …
K = 6 …
K = 7 …
K = 8 …
K = 9 (5/6)^8 * 1/6
K = 10 (5/6)^9
≈5.03

6. Question 27 Section 7.4
What is the variance of the number of heads that come up when a fair coin is flipped 10 times?

7. Question 27 Section 7.4
What is the variance of the number of heads that come up when a fair coin is flipped 10 times?
V(X) = E(X^2) – E(X)^2
P(X = k) = C(10, k)*( (0.5)k*(0.5)10-k)
E(X) = n*p = 10 * 0.5 = 5
E(X2) = 27.5
5/2

8. Exercise 3 Section 9.1
For each of these relations on the set{1,2,3,4}, decide whether it is reflexive, whether it is symmetric, whether it is antisymmetric, and whether it is transitive.
a) {(2,2), (2,3), (2,4), (3,2), (3,3), (3,4)}
b) {(1,1), (1,2), (2,1), (2,2), (3,3), (4,4)}
c) {(2,4), (4,2)}
d) {(1,2), (2,3), (3,4)}
e) {(1,1), (2,2), (3,3),(4,4)}
f) {(1,3), (1,4), (2,3), (2,4), (3,1), (3,4)}

9. Exercise 3 Section 9.1
For each of these relations on the set{1,2,3,4}, decide whether it is reflexive, whether it is symmetric, whether it is antisymmetric, and whether it is transitive.
a) {(2,2), (2,3), (2,4), (3,2), (3,3), (3,4)}
Transitive
b) {(1,1), (1,2), (2,1), (2,2), (3,3), (4,4)}
Reflexive, symmetric, transitive
c) {(2,4), (4,2)}
Symmetric
d) {(1,2), (2,3), (3,4)}
Antisymmetric
e) {(1,1), (2,2), (3,3), (4,4)}
Reflexive, symmetric, antisymmetric, transitive
f) {(1,3), (1,4), (2,3), (2,4), (3,1), (3,4)}
None

10. Exercise 33 Section 9.1
Let R be the relation on the set of people consisting of pairs (a, b), where a is a parent of b. Let S be the relation on the set of people consisting of pairs (a, b), where a and b are siblings (brothers or sisters). What are S◦R And R◦S?

11. Let R be a relation from a set A to a set B and S a relation from B to a set C. The composite Of R and S is the relation consisting of ordered pairs (a, c), where a ∈ A, c ∈ C, and for which there exists an element b ∈ B such that(a, b) ∈ R and (b, c) ∈ S. We denote the composite of R and S by S◦R

12. Exercise 33 Section 9.1
Let R be the relation on the set of people consisting of pairs (a, b), where a is a parent of b. Let S be the relation on the set of people consisting of pairs (a, b), where a and b are siblings (brothers or sisters). What are S◦R And R◦S?
S◦R={(a, b) | a is a parent of b and b has a sibling}
R◦S={(a, b)|a is an aunt or uncle of b}

13. Exercise 1 Section 9.3
Represent each of these relations on {1,2,3} with a matrix (with the elements of this set listed in increasing order).
a) {(1,1), (1,2), (1,3)}
b) {(1,2), (2,1), (2,2), (3,3)}
c) {(1,1), (1,2), (1,3), (2,2), (2,3), (3,3)}
d) {(1,3), (3,1)}

14. Exercise 1 Section 9.3
Represent each of these relations on {1,2,3} with a matrix (with the elements of this set listed in increasing order).
a) {(1,1), (1,2), (1,3)}
b) {(1,2), (2,1), (2,2), (3,3)}
c) {(1,1), (1,2), (1,3), (2,2), (2,3), (3,3)}
d) {(1,3), (3,1)}
111
000 b)
010
110
001 c)
011 d)
100

15. Exercise 18
Draw the directed graphs representing each of the relations from Exercise 1
a) {(1,1), (1,2), (1,3)}
b) {(1,2), (2,1), (2,2), (3,3)}
c) {(1,1), (1,2), (1,3), (2,2), (2,3), (3,3)}
d) {(1,3), (3,1)}