Question

In: Statistics and Probability

Consider a family with a mother, father, and two children. Consider the following set of events:...

Consider a family with a mother, father, and two children. Consider the following set of events: A1 = {mother has influenza} A2 = {father has influenza} A3 = {first child has influenza} A4 = {second child has influenza} B = {at least one child has influenza} C = {at least one parent has influenza} D = {at least one family member has influenza}

1. What does A1 ∪ A2 mean? Which answer is correct? Please provide explanation

a. Both the mother and the father have influenza

b. Neither the mother nor the father has influenza

c. Either the mother or the father has influenza

d. All the family members have influenza

2. Are A3 and A4 mutually exclusive events? Which answer is correct? Please provide explanation

a. Yes

b. No

c. Maybe

3. Express D in terms of B and C. Which answer is correct? Please provide explanation

a. D = B or C

b. D does not equal B or C

c. D = B & C

Suppose an influenza epidemic strikes a city. In 10% of families, the mother has influenza. In 10% of families, the father has influenza. In 2% of families, both the mother and father have influenza. Furthermore, suppose each child has a 20% chance of contracting influenza and there is a 10% chance both children become diseased.

4. Considering the set of events from Question 5, are A1 and A2 independent? Which answer is correct? Please provide an explanation.

a. Maybe

b. Yes

c. No

5. What is the probability that at least one child will get influenza? Which answer is correct? There might be slight rounding differences. Please provide an explanation.

a. 0.2

b. 0.5

c. 0.3

d. 0.1

6. Based on your result from Part (a), what is the conditional probability that the father has influenza given the mother has influenza? Which answer is correct? There might be slight rounding differences. Please provide an explanation.

a. 0.5

b. 0.2

c. 0.3

d. 0.1

Solutions

Expert Solution

(1)

Correct option:

c.     Either the mother or the father has influenza

Explanation:

Given:

A1 = {mother has influenza}

A2 = {father has influenza}

So,

implies either mother has influenza OR father has influenza

(2)

Correct option:

b.    No

Explanation:

Given:

A3 = {first child has influenza}

A4 = {second child has influenza}

Since A3 and A4 can occur together, A3 and A4 are not mutually exclusive.

(3)

Correct option:

a.   D = B or C

Explanation:
Given:

B = {at least one child has influenza}

C = {at least one parent has influenza}

D = {at least one family member has influenza}

So,

D = B or C

(4)

Correct option:

No

Explanation:

P(A1) = P{mother has influenza} = 0.10

P(A2) = P{father has influenza} = 0.10

P(A1 AND A2) = P( both the mother and father have influenza ) = 0.02

Since P(A1 AND A2) =0.02 P(A1) X P(A2) = 0.10 X 0.10 = 0.01, A1 and A2 are not independent.

(5)

Correct option:

c.   0.3

Explanation:

the probability that at least one child will get influenza = 0.2 + 0.2 - 0.1 = 0.3

(6)

Correct option:

b.   0.2

Explanation:

P(A2/A1) = P(A1 AND A2)/P(A1)

               = 0.02/0.10

               = 0.2


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