Math 251 Science Supplement for Exam 2
1. Suppose that an influenza epidemic strikes a city and suppose that data is collected from families that
have a father, a mother, and two children. Suppose that in 10% of the families the mother gets influenza,
in 12% of the families the father gets influenza, and in 5% of the families both parents get influenza.
(a) Within a family what is the probability that neither parent gets the influenza?
(b) Within a family what is the probability that exactly one parent gets influenza?
(c) Within a family what is the probability that at least one parent gets influenza?
(d) Within a family what is the probability that at most one parent gets influenza?
(e) Within a family what is the conditional probability that the father gets influenza given that the mother
gets influenza?
(f) Within a family what is the conditional probability that the mother gets influenza given that the father
gets influenza?
(g) Is the father's getting of influenza independent of the mother's getting of influenza?
2. Suppose that an influenza epidemic strikes a city and suppose that data is collected from families that
have a father, a mother, and two children. Suppose that each child has a 25% chance of getting influenza
and that in 14% of the families both children get influenza.
(a) Within a family what is the probability that neither child gets the influenza?
(b) Within a family what is the probability that exactly one child gets influenza?
(c) Within a family what is the probability that at least one child gets influenza?
(d) Within a family what is the probability that at most one child gets influenza?
(e) Within a family what is the conditional probability that one child gets influenza given that the other
child gets influenza?
(f) Is one child's getting of influenza independent of the other child's getting of influenza?
3. Suppose that an influenza epidemic strikes a city and suppose that data is collected from families that
have a father, a mother, and two children. Suppose that each child has a 25% chance of getting influenza.
Furthermore suppose that 40% of the children whose siblings get influenza also get influenza. i.e. The
conditional probability that a child gets influenza given that his/her sibling gets influenza is 40%. In
addition suppose that 80% of the children whose siblings don't get influenza also don't get influenza.
(a) Within a family what is the probability that neither child gets the influenza?
(b) Within a family what is the probability that exactly one child gets influenza?
(c) Within a family what is the probability that at least one child gets influenza?
(d) Within a family what is the probability that at most one child gets influenza?
(e) Within a family what is the probability that both children get influenza?
(f) Is one child's getting of influenza independent of the other child's getting of influenza?
Math 251 Science Supplement for Exam 2
page two
4. Suppose that the following table gives the number of cases of Alzheimer's disease per 100 people.
Age group
Males
Females
65 through 69
70 through 74
75 through 79
80 through 84
85 through 89
1.8
0.0
5.1
8.8
36.0
0.0
2.4
2.5
8.0
28.0
Suppose that three people are randomly selected: an 85-year-old woman, an 85-year-old man, and a
79-year-old man.
(a) What is the probability that none of the three have Alzheimer's disease?
(b) What is the probability that exactly one of the three has Alzheimer's disease?
(c) What is the probability that exactly two of the three have Alzheimer's disease?
(d) What is the probability that all three have Alzheimer's disease?
(e) What is the probability that at least one of the three has Alzheimer's disease?
(f) What is the probability that at least two of the three have Alzheimer's disease?
(g) What is the probability that at most one of the three has Alzheimer's disease?
(h) What is the probability that at most two of the three have Alzheimer's disease?
(i) What is the probability that both of the men have Alzheimer's disease?
(j) What is the probability that both of the older people have Alzheimer's disease?
(k) Given that exactly one of the three has Alzheimer's disease what is the conditional probability that
the afflicted person is a man?
(l) Given that exactly one of the three has Alzheimer's disease what is the conditional probability that
the afflicted person is over 80 years old?
(m) Given that exactly two of the three have Alzheimer's disease what is the conditional probability that
both are men?
(n) Given that exactly two of the three have Alzheimer's disease what is the conditional probability that
both are over 80 years old?
5. Using data from the preceeding table assume that a married couple is randomly selected. Suppose that
both members of the couple are over 85 years old. Suppose that for couples in that age group the
probability of both spouses having Alzheimer's disease is 0.1.
(a) What is the probability that neither spouse has Alzheimer's disease?
(b) What is the probability that exactly one spouse has Alzheimer's disease?
(c) What is the probability that both spouses have Alzheimer's disease?
(d) What is the probability that at least one spouse has Alzheimer's disease?
(e) What is the probability that at most one spouse has Alzheimer's disease?
(f) Given that the wife has Alzheimer's disease what is the conditional probability that the husband has
Alzheimer's disease?
(g) Given that the husband has Alzheimer's disease what is the conditional probability that the wife also
has Alzheimer's disease?
Math 251 Science Supplement for Exam 2
page three
6. Suppose that a test has been developed to detect a certain type of cancer and suppose that the test
correctly identifies 95% of the cancerous people as cancerous and correctly identifies 80% of the
noncancerous people as noncancerous. Suppose that 0. 2% of the population has this type of cancer. i.e.
The sensitivity of the test is 95%, the specificity of the test is 80%, and the prevalence of this type of cancer
in the population is 0. 2% .
(a) Given that the test indicates that a person has cancer what is the conditional probability that the
person actually has cancer? i.e. What is the predictive value positive?
(b) Given that the test indicates that a person does not have cancer what is the conditional probability
that the person actually does not have cancer? i.e. What is the predictive value negative?
7. Suppose that among heterosexual married couples the probability that the woman is hypertensive is 0.2
and the probability that the man is hypertensive is 0.3. Assume that the woman's blood pressure is
independent of the man's blood pressure. Let X be a random variable representing the number of
hypertensive adults in the couple.
(a) What is the probability distribution table for X?
(b) What is its expected value?
(c) What is its variance?
8. Suppose that among heterosexual married couples the probability that the woman is hypertensive is w
and the probability that the man is hypertensive is h. Assume that the woman's blood pressure is
independent of the man's blood pressure. Let X be a random variable representing the number of
hypertensive adults in the couple.
(a) What is the probability distribution table for X?
(b) What is its expected value?
(c) What is its variance?
9. Evaluate by hand:
(a) 7 !
(b) 8 C 3
(c)
12 
 
4
10. Show that the two formulas given for the variance of a random variable are equivalent. i.e. Show that
n
2
 xi    Pr X
i 1
n

 xi    xi2  Pr  X  xi    2
 i 1

Partial Answers to Math 251 Science Supplement for Exam 2
1. (a) 0.83
(e) 0.5
(b) 0.12
(f) 5/12
(c) 0.17
(d) 0.95
(g) no
2. (a) 0.64
(b) 0.22
(c) 0.36
(d) 0.86
(e) 0.56
(f) no
3. (a) 0.6
(b) 0.3
(c) 0.4
(d) 0.9
(e) 0.1
(f) no
4. (a) 0.4372992 (b) 0.4395424
(c) 0.1180076
(g) 0.8768416
(h) 0.9948592
(i) 0.01836
(m) 0.112019904 (n) 0.810618977
(d) 0.0051408
(j) 0.1008
5. (a) 0.46
(e) 0.9
(d) 0.54
(b) 0.44
(f) 5/14
(c) 0.1
(g) 5/18
6. (a) 19/2015  0.00942928
(b) 7984/7985  0.999874765
7.
(a)
(b) 0.5 adults
Pr ( X = 0 ) = 0.56
Pr ( X = 1 ) = 0.38
Pr ( X = 2 ) = 0.06
8.
(a)
Pr ( X = 0 ) = (1-w)(1-h)
Pr ( X = 1 ) = w + h - 2wh
Pr ( X = 2 ) = wh
(e) 0.5627008
(f) 0.1231484
(k) 0.613095801 (l) 0.946533486
(c) 0.37 adults 2
(b) w + h adults
(c) w(1-w) + h(1-h) adults 2
9. (a) 5040
(b) 56
(c) 495
n
10.
 x
i 1
n
   Pr  X  xi    xi2  2xi   2 Pr  X  xi 
n
2
i
i 1
n
n
i 1
i 1
  xi2 Pr  X  xi   2   xi Pr  X  xi    2  Pr  X  xi 
i 1


  xi2  Pr  X  xi   2 2   2
 i 1

n


  xi2  Pr  X  xi    2
 i 1

n
A More Detailed Answer to Math 251 Science Supplement for Exam 2
8.(c)
xi
Pr X  xi 
0
xi   2
Pr X  xi xi   
1    hw
xi  

2
1
  2hw
1 
1  2   2
2
hw
2
4  4   2
 2   3   2 hw
  2 2   3  2hw  4hw  2 2 hw
4hw  4hw   2 hw
2
  2    2hw
  2    2hw =  h  w  h  w  2hw =  h 2  2hw  w 2  h  w  2hw
2
= hh  ww
2
2
=
h1  h  w1  w