1. (25 points) You are an art agent auctioning off a classic 1844 landscape painting by Thomas Cole at a reputable auction house. The current auction estimate for your painting is $800,000. The seller's reservation price is $600,000, which is the price below which the painting would not be sold (i.e., the seller would refuse to sell below $600,000). The seller's reservation price is not revealed to the potential buyers.

Each bidder submits a single sealed bid for the painting, which means they submit their bid without any knowledge of the bids submitted by the other bidders. The winner, if the reservation price is met, is the bidder submitting the highest bid. The price paid, if the reservation price is met, is the highest bid. If the reservation price is not met (i.e., all bids are below the reservation price), there is no sale.

The actual selling price will depend on two factors:

1. The number of bidders that participate in the auction (more is better!)
2. The individual bids submitted by these bidders

Many Thomas Cole paintings have been sold by the auction house, and based on this data, you have determined that the number of bidders participating in the auction has a distribution described as follows:

Number of Bidders	Probability
1	.1
2	.30
3	.35
4	.15
5	.10

Every bidder that participates knows the current auction estimate ($800,000), but they submit a bid that is based on what they think it is worth. The bid each one submits can be thought of as a multiplicative factor of the auction estimate. A factor of 1.0 means the bidder thinks the auction estimate is spot on. A factor of .9 means the bidder thinks it is worth 10% less (.9 x $800000 = $720,000); a factor of 1.1 means the bidder thinks it is worth 10% more (1.1 x 800,000 = $880,000), and so on and so on. Analysis of previous Thomas Cole auction bids shows that the multiplicative factor an individual bidder applies to the auction estimate comes from a normal distribution with a mean of 1.0 and a standard deviation of 0.25.

Using 10,000 trials, answer the following questions (in your booklet):

1. If the painting sells (i.e., exclude "no sales"), what is the average selling price?
2. What is the probability the painting does not sell (i.e., the reservation price is not met)?
3. What is the probability the painting sells for more than $950,000 (include "no sales" in your analysis)?

Hint: First create five bids, then decide how many of them (1, 2, 3, 4, or 5) are actual bids. Determine the winning bid based on the actual bids.