6.54 Auctioning paintings A collector is interested in two paintings by the same artist available at an auction. She plans to bid $3,000 for the first painting auctioned off and

$2,000 for the second. She estimates that the probability she will win the bid for the first painting (and spend

$3,000) is 30%. If she gets that first painting, she thinks there is an 80% chance she will also win the bid for the second painting (and spend another $2,000). If she does not win the bid for the first painting, she thinks there is only a 10% chance she will win the bid for the second painting. Let X denote the random variable denoting the amount of money the collector will spend at the auction.

a. List the sample space of all possible outcomes of winning or losing the bids for the two paintings. (Draw a tree diagram.)

b. Are the events of winning the bid for the first painting and winning the bid for the second painting independent events? Explain.

c. Find the probability of each outcome in the sample space. (Use the tree diagram.)

d. Find the probability distribution of X.

e. Find the mean of X.