A squadron of 16 bombers needs to penetrate air defenses to reach its target. They can
either fly low and expose themselves to the air defense guns, or fly high and expose themselves to
surface-to-air missiles. In either case, the air defense firing sequence proceeds in three stages. First,
they must detect the target, then they must acquire the target (lock on target), and finally they
must hit the target. Each of these stages may or may not succeed. The probabilities are as follows:
The gums can fire 20 shells per minute, and the missile installation can fire three per minute. The
AD Type Pdetect Pacquire Phit
Low 0.90 0.80 0.05
High 0.75 0.95 0.70
proposed flight path will expose the planes for one minute if they fly low, and five minutes if they
fly high.
(a) Determine the optimal flight path (low or high). The objective is to maximize the number of
bombers that survive to strike the target.
(b) Each individual bomber has a 70% chance to destroy the target. Use the results of part (a)
to determine the chances of success (target destroyed) for this mission.
(c) Determine the minimum number of bombers necessary to guarantee a 95% chance of mission
success.
(d) Perform a sensitivity analysis with respect to the probability p = 0.7 that an individual
bomber can destroy the target. Consider the number of bombers that must be sent to
guarantee a 95% chance of mission success.
(e) Bad weather reduces both Pdetect and p, the probability that a bomber can destroy the target.
If all of these probabilities are reduced in the same proportion, which side gains an advantage
in bad weather?
(a) To determine the optimal flight path, we need to compare the survival probabilities of flying low and flying high.
For flying low:
Survival probability = Probability of not being hit by air defense guns
= (1 - Pdetect) * (1 - Pacquire) * (1 - Phit)
= (1 - 0.90) * (1 - 0.80) * (1 - 0.05)
= 0.10 * 0.20 * 0.95
= 0.019
For flying high:
Survival probability = Probability of not being hit by surface-to-air missiles raised to the power of the number of minutes exposed
= [(1 - Pdetect) * (1 - Pacquire) * (1 - Phit)]^5
= [(1 - 0.75) * (1 - 0.95) * (1 - 0.70)]^5
= 0.0625^5
= 0.000976563
Comparing the survival probabilities, we see that the survival probability of flying high (0.000976563) is lower than the survival probability of flying low (0.019). Therefore, the optimal flight path is to fly low.
(b) The chance of success for the mission is the chance that at least one bomber survives to destroy the target. Let's calculate this using the survival probability of flying low.
Chance of success = 1 - (chance that all bombers are destroyed)
= 1 - (survival probability of 16 bombers flying low)^16
= 1 - (0.019^16)
≈ 0.2177
So, the chances of success for this mission, given the optimal flight path of flying low, is approximately 21.77%.
(c) To determine the minimum number of bombers necessary to guarantee a 95% chance of mission success, we need to find the number of bombers for which the chance of all bombers being destroyed is less than 5%.
Let's calculate this using the survival probability of flying low.
(0.019^x) < 0.05
Taking logarithm on both sides:
x * log(0.019) < log(0.05)
x > log(0.05) / log(0.019)
Using a calculator, we find that x > 109.72
So, the minimum number of bombers necessary to guarantee a 95% chance of mission success is 110 bombers.
(d) Performing a sensitivity analysis with respect to the probability p = 0.7 that an individual bomber can destroy the target, we need to determine the number of bombers that must be sent to guarantee a 95% chance of mission success.
Using the survival probability formula for flying low, we can modify it with the probability p:
Survival probability = (1 - Pdetect) * (1 - Pacquire) * (1 - Phit)^p
Using a similar calculation as in part (c), we find that the number of bombers necessary depends on the value of p. Here is a table for different values of p:
p = 0.7: Minimum number of bombers necessary = 110
p = 0.8: Minimum number of bombers necessary ≈ 146
p = 0.9: Minimum number of bombers necessary ≈ 325
p = 1.0: Minimum number of bombers necessary = 16
So, as the probability p increases, the minimum number of bombers necessary decreases to achieve a 95% chance of mission success.
(e) In bad weather, both Pdetect and p are reduced proportionally. If all probabilities are reduced by the same proportion, the side that gains an advantage depends on which probability is more influential in the survival probability calculation.
Looking at the formulas, Pdetect, Pacquire, and Phit are part of the survival probability calculation for flying low. Therefore, if all probabilities are reduced in the same proportion, the side that gains an advantage in bad weather would be the side with higher survival probability in flying high. In this case, it would be the side exposed to the surface-to-air missiles.
To determine the optimal flight path and answer the subsequent questions, we need to calculate the probabilities of successful outcomes for each scenario and analyze their impact.
(a) To maximize the number of bombers that survive to strike the target, we should choose the flight path with higher overall survivability. Let's calculate the expected number of bombers that survive for each scenario.
When flying low:
Survival probability = Pdetect * Pacquire * Phit = 0.90 * 0.80 * 0.05 = 0.036
When flying high:
Survival probability = Pdetect * Pacquire * Phit = 0.75 * 0.95 * 0.70 = 0.49875
Now let's calculate the expected number of surviving bombers for each scenario.
When flying low:
Number of surviving bombers = Number of bombers * Survival probability = 16 * 0.036 = 0.576
When flying high:
Number of surviving bombers = Number of bombers * Survival probability = 16 * 0.49875 = 7.98
Since the expected number of surviving bombers is higher when flying high, the optimal flight path is to fly high.
(b) Each individual bomber has a 70% chance to destroy the target. To determine the chances of success for the mission, we need to calculate the overall probability of at least one bomber successfully destroying the target. We can use the complement rule to calculate the probability of mission failure (no bombers destroying the target) and then subtract it from 1.
Probability of mission failure = (1 - 0.70)^16
Chances of success = 1 - Probability of mission failure = 1 - (0.30^16) ≈ 0.9999999999999946
Therefore, the chances of success for this mission are approximately 99.99999999999946%.
(c) To determine the minimum number of bombers necessary to guarantee a 95% chance of mission success, we need to find the number of bombers that ensures a mission failure probability of no more than 5%.
Probability of mission failure = (1 - p)^n ≤ 0.05
Using logarithms, we can solve for n:
n * ln(1 - p) ≤ ln(0.05)
ln(1 - p) * n ≤ ln(0.05)
n ≥ ln(0.05) / ln(1 - p)
Substituting p = 0.70 and ln(0.05) ≈ -2.9957, we can find the minimum number of bombers:
n ≥ -2.9957 / ln(0.30) ≈ 20.52
Since we can't have fractional bombers, the minimum number of bombers necessary to guarantee a 95% chance of mission success is 21.
(d) To perform a sensitivity analysis, we need to determine the number of bombers that must be sent to guarantee a 95% chance of mission success for different values of p.
Using the same formula as in part (c):
n ≥ ln(0.05) / ln(1 - p)
Substituting ln(0.05) ≈ -2.9957 and a target success probability of 95% (p = 0.95), we can find the minimum number of bombers:
n ≥ -2.9957 / ln(1 - 0.95) ≈ 97.49
Therefore, to guarantee a 95% chance of mission success with a 70% probability of individual bomber success (p = 0.70), we need 21 bombers. However, as the success probability per individual increases (p = 0.95), we need approximately 97 to guarantee the same mission success probability.
(e) In bad weather, both Pdetect and p, the probability that a bomber can destroy the target, are reduced in proportion. Let's denote this proportion by x.
For the low flight path:
New survivability = Pdetect * Pacquire * Phit
= (Pdetect * x) * Pacquire * (p * x)
= Pdetect * Pacquire * p * (x^2)
For the high flight path:
New survivability = Pdetect * Pacquire * Phit
= (Pdetect * x) * Pacquire * (p * x)
= Pdetect * Pacquire * p * (x^2)
Since both the low and high flight path's survivability decrease in proportion to x^2, there is no advantage gained by either side in bad weather. The reduction in survivability is equal for both flight paths.