Please provide solutions to the following Questions and show all necessary work.

Question

Please provide solutions to the following Questions and show all necessary work.

9. Suppose an influenza epidemic strikes a city. In 10% of families the mother has influenza; in 10% the father has influenza, and in 2% both parents have influenza. Suppose also that there is a 20% chance of each child will get influenza, whereas in 10% of two-child families both children get influenza. Consider a family with a mother, a father, and two children from that city. {Use probability theory to answer the following questions, not just your own intuition.) (a) What is the probability of only one parent in the family having influenza. [HINT: Use a Venn diagram.] (b) Does the chance of the mother having the disease depend on whether the father also has it, or the two events are completely independent (c) What is the probability of at least one child getting influenza? (d) What is the probability that neither of the parents will get influenza given that both children have the disease. (e) What is the probability that at least one person in the family will get influenza? HINT: For parts (d) and (e) ezpress the respective events as a combination of the events that parents and/or children are getting the disease using union, intersection and com- pliments of events. Can you provide a final number for the respective probabilities? Explain.] (f) [BONUS] Suppose that the event that any of the parents will get influenza is independent of the event that any of the children also getting it. Use this information to calculate parts (d) and (e)

Explanation / Answer

A) P(one parent having influenza) = 0.1 * 0.9 + 0.1 * 0.9 = 0.18

B) P(mother and father having disease ) = 0.02

P(mother having disease) * P(father having disease) = 0.1 * 0.1 = 0.01

As these two probabilities are not equal so the two events are not independent.

C) P(at least one child has influenza) = 0.1 * 0.9 + 0.2 = 0.29

D) P(neither of the parents have influenza | both children have disease) = (1 - 0.2) * 0.1/0.1 = 0.8

E) P(at least one person in the family will get influenza) = 1 - (no one has influenza) = 1 - (0.9 * 0.9 * 0.9 * 0.8) = 0.4168

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