Question

A radar complex consists of eight units that operate independently. The probability
that a unit detects an incoming missile is 0.9. Find the probability that an incoming missile will
not be detected by any unit.

Answer 1

Hint: In this problem, first we have to find the probability of not detecting the missile. Since the
probability of detecting the missile is given, so by subtracting that value from 1 we get the
probability of not detecting the missile. In the question it is mentioned that the radar complex
contains eight operator units. So we need to calculate the probability of not detecting the missile
for eight units. For that we need to use the formula $P\left( {{A_1} \cap {A_2} \cap \ldots \cap
{A_n}} \right)$.
It is given that the probability of detecting the missile = 0.9
Now, we have to subtract this value from 1. This will give the probability of not detecting the
missile.
Probability of not detecting the missile = 1 $ - $ the probability of detecting the missile
$ = 1 - 0.9$
$ = 0.1$
Since, there are eight units that operate independently so we have to find the probability that the
incoming missiles will not be detected by any unit.
Probability for the incoming missile will not be detected by any unit is $P\left( {{A_1} \cap
{A_2} \cap \ldots \cap {A_8}} \right)$.
$\begin{array}{c}P\left( {{A_1} \cap {A_2} \cap \ldots \cap {A_8}} \right) = \prod\limits_{i =
1}^8 {P\left( {{A_i}} \right)} \\ = {\left( {0.1} \right)^8}\\ = {10^{ - 8}}\end{array}$
Hence, the probability for the incoming missile will not be detected by any unit is ${10^{ - 8}}$.
Note: Here after calculating the probability of not detecting the missie for one unit, it is also important to calculate the summation of all the 8 units.Do not leave the answer after calculating the probability of one unit