DA4 is a television station that has 25 thirty-second advertising slots during each evening. It is early January and the station is selling advertising for Sunday, March 24. They could sell all of the slots right now for $4,000 each, but, because on this particular Sunday the station is televising the Oscar ceremonies, there will be an opportunity to sell slots during the week right before March 24 for a price of $10,000. For now, assume that a slot not sold in advance and not sold during the last week is worthless to DA4. To help make this decision, the salesforce has created the following probability distribution for last-minute sales: Number of Slots, x Probability Exactly x Slots Are Sold
8 0.00
9 0.05
10 0.10
11 0.15
12 0.20
13 0.10
14 0.10
15 0.10
16 0.10
17 0.05
18 0.05
19 0.00
a. How many slots should DA4 sell in advance?
b. In practice, there are companies willing to place standby advertising messages: if there is an empty slot available (i. e., this slot was not sold either in advance or during the last week), the standby message is placed into this slot. Since there is no guarantee that such a slot will be available, standby messages can be placed at a much lower cost. Now suppose that if a slot is not sold in advance and not sold during the last week, it will be used for a standby promotional message that costs advertisers $2,500. Now how many slots should DA4 sell in advance?
c. Suppose DA4 chooses a booking limit of 12 slots on advanced sales. In this case, what is the probability there will be slots left over for stand-by messages?