a missile protection system consists of n radar sets operating independently each with probability .9 of defecting an aircraft entering a specified zone.all radar sets cover the same zone.if an airplane enters the same, find the probability that it will be detected if n=2 and n=4
To find the probability that an airplane will be detected by a missile protection system, we need to consider the probability of at least one radar set detecting the aircraft.
When n=2:
The probability that both radar sets fail to detect the aircraft is (0.1)² = 0.01.
Therefore, the probability that at least one radar set detects the aircraft is 1 - 0.01 = 0.99.
When n=4:
The probability that all four radar sets fail to detect the aircraft is (0.1)⁴ = 0.0001.
Therefore, the probability that at least one radar set detects the aircraft is 1 - 0.0001 = 0.9999.
Thus, if n=2, the probability that the airplane will be detected is 0.99, and if n=4, the probability is 0.9999.
To find the probability that an airplane will be detected by a missile protection system with n radar sets, we need to consider the probability of all radar sets failing to detect the airplane.
Let's start with the case where n=2:
Probability of both radar sets failing to detect the airplane = Probability of the first radar set failing * Probability of the second radar set failing
Since each radar set has a probability of 0.9 of detecting the airplane, the probability of one radar set failing to detect the airplane is 1 - 0.9 = 0.1.
Therefore, the probability of both radar sets failing to detect the airplane is 0.1 * 0.1 = 0.01.
The probability that the airplane will be detected is then 1 - 0.01 = 0.99.
Now let's consider the case where n=4:
Probability of all radar sets failing to detect the airplane = Probability of the first radar set failing * Probability of the second radar set failing * Probability of the third radar set failing * Probability of the fourth radar set failing
Again, each radar set has a probability of 0.9 of detecting the airplane, so the probability of one radar set failing to detect the airplane is 1 - 0.9 = 0.1.
Therefore, the probability of all four radar sets failing to detect the airplane is 0.1 * 0.1 * 0.1 * 0.1 = 0.0001.
The probability that the airplane will be detected is then 1 - 0.0001 = 0.9999.
So, if n=2, the probability that the airplane will be detected is 0.99, and if n=4, the probability is 0.9999.