six questions on decision making and propability

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1. the smt company producer of a new product isn't sure what the ultimate sales price for the product will be and isn't sure of the price the market will establish for the product. the probability distribution describing the possible combinations of demand and price over the 1st quarter of sales is shown following. you support this product, so you are interested in estimating the demand and ultimate sales price of the product in the 1stquarter.
demand (d)

price per unit (p) 2000 2500 3000 3500
$20 0.05 0.05 0.03 0.15 0.28
$25 0.05 0.06 0.10 0.05 0.26
$30 0.08 0.10 0.04 0.03 0.25
$35 0.10 0.05 0.03 0.03 0.21
0.28 0.26 0.20 0.26

a. find the expected price and the expected demand level for the upcoming quarter.
b. what is the probability that the price of this product will be above its mean (expected value) in the upcoming quarter?
c. what is the probability that the demand for this product will be below its mean (expected value) in the upcoming quarter?
d.. what is the probability that the demand of this product exceeds 2500 units in the upcoming quarter, given that its price is less than $30?
e. what is the probability that the price is below $30 and the demand is above 3000 units in the upcoming quarter?
f. the correlation between demand and price based on the data above is –0.3553. is this consistent with your expectations? explain.

2. delays of shipments at border crossings are a high contributor to costs as well as a contributor to the risk of late or lost shipments. a trade association take a sample of information on the length of shipment delays at a number of maritime ports and the presence of active compliance with wto “best practices customs procedures” at those ports to see if the wto best practices were making an impact where they had been fully implemented. the table below shows some of the data that was collected. the number in each cell is the number of border crossings for that combination of compliance status and days delayed.
number of days crossing is delayed
wto “best practice” compliance? 1 3 5 or more
yes 60 110 30 200
no 40 215 45 300
100 325 75 500

a. what proportion of delays lasted more than one day?
b. what proportion of delays lasted only a single day at a wto-compliant port?
c. given that a port is wto compliant, what proportion of delays lasted a single day?
d. given that the delay is more than one day, what proportion of delays were at non-compliant ports? that is, what proportion of delays in this sample that are more than one day are at non-compliant ports in this study?
e. given that the delay is 5 days or more, what proportion of these delays are at non-compliant ports?
f. based on the data from this survey, is there evidence that wto best customs practices are reducing border delays at maritime ports? explain.

3. concerns have been coming in from strategic partners that payments for their shipments are being delayed due to minor “holds” in the oracle payment system and some payments are as much as 90 days or more later than contracted. a sample of oracle “holds” on payments to strategic partners is taken to see how late they are. the small table below records the results of this sampling.
days late (beyond contracted time) proportion of payments
30 .55
60 .25
90 .15
more than 90 .05

a. what is the forecasted delay for a payment with a “hold”?
b. what is the probability that a payment on “hold” is more than 30 days late?
c. given that a payment is more than 30 days late, what is the forecasted delay?

4. hire car company routinely “over book” their fleet of cars because they know that not all those who have reservations show up to use the car they reserved and customers usually don’t call, either. they just don’t show up (missed flight, change in plans, didn’t know it was booked, etc.). one rental agency at a particular airport says that once a customer books a car, there is a 90% chance they will show up that day to take the car. suppose that all the renters are independent and all have a 90% chance of showing up. if the hire company has 100 cars to rent, how many reservations should they book if they want to be 95% sure that they won’t have more than 100 people show up?

5. for a vendor transaction to be completed properly, the good or service has to (1) be received on time, (2) it has to be on spec, (3) the payment has to be sent on time to the supplier, and (4) the payment amount has to be correct. suppose each of these steps is done correctly with a reliability of 0,98.
a. in 10,000 “transactions” with suppliers, how many have no problems at all?
b. in 10,000 transactions with suppliers, how many have at least one step go wrong?
c. in 10,000 transactions with suppliers, how many have exactly one step go wrong?
d. in 10,000 transactions with suppliers, how many have a problem with all four steps?

6. delay times for shipments from company mmm are normally distributed with a mean of 6 days and a standard deviation of 2 days. delay times for shipments from company mm are normally distributed with a mean of 9 days and a standard deviation of 6 days.
a. if you order a shipment from each vendor, what is the probability that each supplier has a delay of 3 days or fewer?
b. what is the probability that a shipment from company mmm is not delayed (that is, a delay of 0 days)?
c. what is the probability that company mm’s shipment is delayed more than 9 days?
d. if orders are placed with both suppliers, what is the probability that both shipments are delayed more than 6 days (that is, mmm’s is more than 6 days delayed and mm’s is more than 6 days delayed)?
e. what is the 80th percentile for delays on shipments from company mmm? that is, what is the number of days delay such that 80% of the shipments from company mmm are that late or less? (this would mean i’m 80% sure i won’t get mmm’s shipment later than that many days beyond contracted time.)
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