1. A submarine is lost on the continental shelf, the bottom is muddy
and detection will be very difficult. The region known to contain the lost sub
has been subdivided into sub regions as shown below with the probabilities the
sub is in each cell (sub region) shown by the bold faced numbers in the cells.
In a half day of search of a cell the probability of detecting the sub given it
is in the cell being searched is .3.
|
2) .23 |
3) .31 |
4) .22 |
a.
Construct
the box Venn diagram used to analyze the first day of search.
b. What is the probability this first search will result in detection?
c. If this first search is unsuccessful what is the probability the sub is in each cell of the region.
d.
If
the probability of detection in cells 1), and 2) is .2 in a half day of search
and in cells 3), and 4) is .15, which cell do you want to searched first and why?
2. A
naval staff is planning an operation in the Gulf. They would like to know the
location of the Iranian’s Kilo class nuclear submarine. It could be in its
homeport with probability .35, or it could be in one of its two normal patrol
regions, or it could be somewhere else. There is a 26% chance it is in patrol
region Alpha and a 24% chance it is in patrol region Gamma. An Unmanned Aerial
Vehicle (UAV) will be launched to check out the port. It has an 80% chance of
detecting the sub if it is in port. A P3-C search of region Alpha has a 50%
chance of detecting the sub if it is there and a 60% chance of detecting it in
a search of region Gamma. It would be unduly provocative to fly more than one
P3-C off the Iranian coast so only one region at a time can be searched.
a.
Should
region Alpha or Gamma be subject to a P3-C search first? You must show
calculations and state a reason.
b. You are to make an analysis
of the plan to have the UAV search the port and to have region Alpha searched
by a P3-C; both these searches will be done on day 1.
i. Draw the box Venn diagram to
be used in this analysis.
ii. What is the probability the
sub is found?
iii. If both the UAV search of the port is made and
region Alpha is searched by a P3-C with no detection, what is the probability
the sub is in each possible place?
3.
A target we want to find has a 10% probability of being in Cell 1, a 70%
probability of being in Cell 2 or could be in Cell 3.When a search of a cell
containing the target is done there is a 60% chance of finding it.
a.
Which
cell should be searched first?
b.
What
is the probability the target is found when the cell identified in part (a) is
searched?
c.
If
the target is not found on this first search, then which cell should be
searched next?
4. The target has the same distribution amongst three cells as in problem 3. Each cell is a region having area of 1,500 square nm. The target is being searched for by an aircraft flying at 200 knots and with a sweep of 5 nm. The probability of detecting the target is given by the random search formula.
a.
If
Cell 2 is searched for 2 hours what is the probability the target is found?
b.
What
is the probability the target is in each of the cells after the search of part
a is completed and the target was not found?
c.
How
long should Cell 2 be searched before starting to search Cell 3?
|
Sub-area 1 |
Sub-area 2 |
|
|
|
|
200 nm |
300 nm |
a.
Suppose
you consider a random search, spending 6 hours searching sub-area 1 and 4 hours
searching sub-area 2. If no detection is made on the search, what would
be the updated probabilities of the ships location?
b.
Suppose
you decide to search sub-area 1 using essentially a random search until the
probabilities that the ship is in the two sub-areas are equal, or the ship is
located. Assuming that the ship is not found, how long do you search
sub-area 1 before changing your tactics?
1.
(b) .165, (c) .201, .275, .260, .263, (d) search cell 1.
2. (b) ii. .41, iii. .119, .220, .407, .254
3. (b) .42. (c) Search Cell 2 again.
4. (a) .515, (b) .206, .381, .413 (c) 1.88 hours.
mailto: tjs@usna.edu
last update: 05 January 2000