1. A1N A2 means that at least one parent has influenza

2. means that both parents have influenza

3. no because both children can have influenza

4. at least one child has influenza

5. first child has influenza

6. C = A1N A2

7. D = B N C

8. mother does not have influenza

9. father does not have influenza

10. notC =notA1 O notA2

11. notD= notB O notC

12. P(A1O A2) =0.02 > P(A1)*P(A2)=0.01

13. P(A1N A2) = P(A1) + P(A2) - P(A1O A2) = .2+.2-.1=.3

    3.31 the probability of both sibling being affected is 1/2*1/2 =1/4

    3.32 The probability that exactly one sibling is affected is 2*1/2*1/2 =0.5

    3.33 The probability that neither sibling is affected 1/2*1/2 =1/4

    3.34 The probability that the younger child is affected should not be influenced by whether or not the older child is affected. Thus, the probability of the younger child being affected remains at \.

    3.35 The events A, B are independent because whether or not a child is affected does not influence the outcome for other children in the family.

    3.36 The probability that both siblings are affected is 1/4*1/4 = 1/16

    3.37 The probability that exactly one sibling is affected = 2*1/4*3/4 = 3/8

    3.38 The probability that neither sibling is affected is 3/4*3/4 = 9/16

    3.39 The probability that both siblings are affected = 0, because the female sibling cannot get the disease

    3.40 The probability that exactly one sibling is affected = \, since only the male sibling can be affected.

    3.41 The probability that neither is affected =1/2*1=1/2

    3.42 P(both affected)=1/2*1/2=1/4

    3.43 P(exactly one affected)= 2*1/2*1/2=1/2

    3.44 P(neither affected) = 1/2*1/2 = \

    3.45 Bayes' theorem

    P(D)=P(R)=P(SL) = 1/3

    from problems 3.31, 3.36, and 3.42 we know following probabilities respectively:

    P(A|D)=1/4 P(A|R)=1/16 P(A|SL)=1/4

    P(D|A)=P(A|D)*P(D)/[P(A|D)*P(D)+P(A|R)*P(R)+P(A|SL)*P(SL)]=4/9

    P(R|A)=P(A|R)*P(R)/[P(A|D)*P(D)+P(A|R)*P(R)+P(A|SL)*P(SL)]=1/9

    P(SL|A)=P(A|SL)*P(SL)/[P(A|D)*P(D)+P(A|R)*P(R)+P(A|SL)*P(SL)]=4/9

    Dominant and Sex-Linked modes of inheritance are the most likely, and recessive is less likely.

    3.46 Let B={exactly one of two male siblings is affected}

    from problems 3.32, 3.37, and 3.43 we know following probabilities:

    P(B|D)=1/2 P(B|R)=3/8 P(B|SL)=1/2

P(D|B)=P(B|D)*P(D)/[P(B|D)*P(D)+P(B|R)*P(R)+P(B|SL)*P(SL)]=4/11

P(R|B)=P(B|R)*P(R)/[P(B|D)*P(D)+P(B|R)*P(R)+P(B|SL)*P(SL)]=3/11 P(SL|B)=P(B|SL)*P(SL)/[P(B|D)*P(D)+P(B|R)*P(R)+P(B|SL)*P(SL)]=4/11

3.47 Let C={both one male and one female sibling are affected}. Note that the sex of the siblings is only relevant for sex-linked disease.

from problems 3.31, 3.36 and 3.39 we know following probabilities:

P(C|D)=1/4 P(C|R)=1/16 P(C|SL)=0

Thus,

P(D|C)=P(C|D)*P(D)/[P(C|D)*P(D)+P(C|R)*P(R)+P(C|SL)*P(SL)]=4/5

P(R|C)=P(C|R)*P(R)/[P(C|D)*P(D)+P(C|R)*P(R)+P(C|SL)*P(SL)]=1/5

P(SL|C)=P(C|SL)*P(SL)/[P(C|D)*P(D)+P(C|R)*P(R)+P(C|SL)*P(SL)]=0

3.48

Let E={male sibling affected, female not affected}.

P(E|D)=1/2*1/2=1/4 P(E|R)=1/4*3/4=3/16 P(E|SL)=1/2*1=1/2

Thus,

P(D|E)=P(E|D)*P(D)/[P(E|D)*P(D)+P(E|R)*P(R)+P(E|SL)*P(SL)]=4/15

P(R|E)=P(E|R)*P(R)/[P(E|D)*P(D)+P(E|R)*P(R)+P(E|SL)*P(SL)]=1/5

P(SL|E)=P(E|SL)*P(SL)/[P(E|D)*P(D)+P(E|R)*P(R)+P(E|SL)*P(SL)]=8/15

3.94 The sensitivity = P(test+|true+) = P(test+|pregnant) = 95/100 =.95

3.95 The specificity =P(test-|true-) = P(test-|not pregnant) = 99/100 = .99

3.96 PV+ = prevalence x sensitivity /{prevalence x sensitivity +(1-prevalence) x (1-specificity)}

= 0.10 x 0.95 /{0.10 x 0.95 + .90 x 0.01}= 0.913

91.3% of women who test positive will actually be pregnant.

3.97 A false negative is a woman who tests negative, but is actually pregnant.

P(false negative) = P(pregnant) x P(test negative|pregnant) =0.1x(1-sensitivity)

A false positive is a woman who tests positive, but is actually not pregnant

P(false positive) = P( not pregnant) x P(test positive | not pregnant) =0.9x(1-specificity)

3.98 The sensitivity = 12/16

3.99 The specificity = 34/46

3.100 Following is the table of sensitivity and specificity according to the cutoff points used

cutoff point sensitivity specificity 1-specificity

0 0 1 0

5 .125 1 0

10 .188 1 0

15 .438 .935 .065

20 .750 .739 .261

25 .938 .391 .609

30 1 0 1

1.101 Plot of the sensitivity vs 1-specificity.

1.102 The only cutoff point is 20.