| No. of Bedrooms | 0 | 1 | 2 | 3 | 4 or more |
| No. of Houses (thousands) |
547 | 5012 | 6100 | 2644 | 557 |
The expected value is the weighted sum of probabilities of the outcomes. Below is a table
showing the necessary calculations:
The sum of the third column gives the expected value: μ = 1.84.
The sum of the sixth column gives the variance: 0.787.
To quickly check your work, remember that the expected value is a sort of average --
your answer should be between 0 and 4. The square root of the variance is the standard
deviation; adding the standard deviation to the average (or subtracting it from the average)
should give you a result between 0 and 4.
x f(x) x*f(x) x-μ (x-μ)2
f(x)*(x-μ)2 0 0.037 0 -1.842 3.392 0.125 1 0.337 0.337 -0.842 0.709 0.239 2 0.410 0.821 0.158 0.025 0.010 3 0.178 0.534 1.158 1.341 0.239 4 0.037 0.150 2.158 4.567 0.175
a) What is the probability of 30 seconds or less between telephone calls?
First, convert 30 seconds to .5 minutes. Then compare to the equation on page 83 to see that the mean time between phone calls is μ=2. Using the formula on page 84 we get: P(x ≤ .5) = 1 - e-.5/2 = .2212 (approximately). So the answer is "Approximately 22%."
| Bonus | No Bonus | Female | 3 | 1 | Male | 7 | 2 |
a) Create a joint probability table for programmer bonuses.
| Bonus | No Bonus | Female | 23% | 8% | 31% | Male | 54% | 15% | 69% | 77% | 23% | 100% |
b) If a programmer got bonus pay, what is the probability that that programmer was female?
This is a conditional probability.
P(female | bonus) = P(female and bonus) / P(bonus) = 23/77 = 30%
No, because 31% of all employees are female. So if there was no bias you would expect about 31% of the bonuses to go to women, which is excactly what happened.
a) What is the optimal decision strategy for Dante?
To generate a decision strategy, first fill in the expected values
of each chance and decision event starting at the right and moving
left. For instance, if market research predicts high demand (chance
node 8), the expected value of building a complex is:
0.85 * 2650 + 0.15 * 650 = 2350
The expected value of selling the rights is only 1150, so the value of
the decision node labeled 5 is 2350. (From this node you have the
option of choosing an expected profit of 2350 or a known profit of
1150; the expected value approach recommends choosing 2350.)
Find the EV of node 6 using the EV of node 9, then use the EV's of
nodes 5 and 6 to compute the EV of node 4:
0.6 * 2350 + 0.4 * 1150 = 1870
It turns out that the expected value of building a complex without
doing market research is 2000 (node 7), so at decision node 3 your
decision strategy should recommend that choice.
The expected value of bidding then turns out to be 1560, which is
better than the expected value of not bidding. So your final answer
would be:
"Dante Development Corporation should bid on the contract. If they
win, they should proceed to build the complex."
The possible payoffs for this strategy are:
Lose contract: -200
The probability of the -200 payoff is 20%.
The probability of winning the contract is 80%, and once this happens
there is a 60% chance of high demand and a 40% chance of moderate
demand. So overall there is a 0.80 * 0.60 = 0.48 or 48% chance of
high demand and a 32% chance of moderate demand.
The risk profile will have three vertical bars, one at x=-200, one at
x=800 and one at x=2800 (please be careful that your graph is to
scale). The vertical bars will have heights 0.20, 0.32 and 0.48
respectively.
Build with high demand: 2800
Bulid with moderate demand: 800
c) If Dante wins the contract but does no market research, does the MiniMax Regret approach recommend building or selling?
The payoff table is as follows:
| High | Moderate | Build | 2800 | 800 | Sell | 1300 | 1300 |
The regret table looks like:
| High | Moderate | Build | 0 | 500 | Sell | 1500 | 0 |
The maximum regret occurs when selling the rights, so minimax regret recommends building.
- Each box of MusicMaker software sold yields $10 profit. Each Computer Cacaphony sold yields $6 profit.
- Merry Melodies has enough space in a warehouse for 50 palettes of software. A palette holds 40 copies of MusicMaker software or 100 copies of Computer Cacaphony.
- Projected demand is for 1800 copies of Merry Melodies and 1500 copies of Computer Cacaphony.
- Merry Melodies' lawyers recommend that at least 40% of the software produced be Computer Cacaphony in order to avoid an antitrust suit.
Variables:
Maximize: 10M + 6C
Subject to:
M = # Music Maker produced
C = # Computer Cacaphony produced
M/40 + C/100 ≤ 50
M ≤ 1800
C ≤ 1500
C ≥ .4(M + C)
M ≤ 0
C ≤ 0
The feasible region is the upper left quadrilateral above (marked). The extreme points of the
feasible region are (0, 1500), (1400, 1500), (approximately) (1579, 1052), and (0,0). Plug these
values for (M, C) in to 10M + 6C to find that the maximum profit occurs at M=1400, C=1500. So the
answer is "Produce 1400 copies of Music Maker and 1500 copies of Computer Cacaphony."
| Activity | Optimistic | Most Probable | Pessimistic |
| A | 3 | 4 | 8 |
| B | 5 | 8 | 11 |
| C | 2 | 4 | 6 |
| D | 4 | 5 | 12 |
Tasks B and C can both be started as soon as A completes, but task D cannot start until B and C are both completed.
a) Develop a project network.
First compute the expected times using the formula:
t = (a + 4m + b)/6
Activity Optimistic Most Probable
Pessimistic Expected Variance A 3 4 8 4.5 0.6944 B 5 8 11 8 1 C 2 4 6 4 0.4444 D 4 5 12 6 1.7778
B|4.5 12.5
8|4.5 12.5
/ \
A |0 4.5 D|12.5 18.5
4.5|0 4.5 6|12.5 18.5
\ /
C|4.5 8.5
4|8.5 12.5
b) What is the expected project completion time?
18.5 days.
c) If the shop obtains the job with a bid that is based on the expected completion time, what is the probability that it will lose money on the job?
The bid is assumed to be equal to parts costs plus labor costs:
$8000 + $400/day * 18.5 days = $15400
d) What bid should the shop make if they wish to have a 90% chance of making a proft?
At this point you need the variance shown in the table above and
computed using the formula:
σ2 = ( (b - a)/6 )2.
As we can see from the network, the critical tasks are A, B and D.
The variance for the entire project is therefore approximately:
0.6944 + 1 + 1.7778 = 3.47.
We have a normal distribution with a mean of 18.5 and a standard
deviation of 1.86. We want a 90% chance of making a profit, so we
need to know a number of days at which we are 90% sure the project is
completed. In other words, at what point on the x-axis is the area
under the curve 90%?
To solve this using a TI 83
Calculator, you would enter InvNorm(.90, 18.5, 1.86).
To solve this using a Z-score you would look up the area to the right
of the mean (0.9 - 0.5 = 0.4) in a table
of z-scores to get the z-score 1.28. You would then solve the
equation:
1.28 = (x - 18.5) / 1.86
for x, to get x = 20.88 days.
We can now calculate a bid based on this number:
$8000 + $400/day * 20.88 days = $16352.
Note: We could have rounded up from 20.88 to 21 and bid $16,400.
However, if we do that and another shop submits a bid for $16,352 we
might lose the contract. On the other hand, rounding down would mean
accepting a greater than 10% chance of taking a loss.