Amy in a New Town

Amy immigrated to a new city, and would like to make friends with her new neighbours.

On any particular day i, she feels shy with probability 1-p (0pp, she goes out and visits the i-th house in her neighborhood. At any house that she visits, either:

(1) someone is at the house and answers the door; this happens with probability q (where 0q

(2) no one is at the house and Amy returns home.

We assume that the collection of all events of the form {Amy stays home on day i} and {Someone is at the i-th house on day i}, for i=1,2,..., are (mutually) independent.

A. What is the probability that Amy gives out a picture on the first day?

B. Fix some integers k and n, with n >= 1 and 0kn. What is the probability that she gave out exactly pictures on the first days?

C. What is the probability that she stayed home on day 1, given that she did not give out a picture on that day?

D. Within the first week (7 days), what is the probability that there were no days in which Amy went out but did not give out a picture of her hometown?

Attachment:

Amy in a New Town Amy immigrated to a new city, and would like to make friends with her new neighbours. On any particular day 2', she feels shy with probability 1 p (0