Question

An anti-aircraft gun can fire four shots at a time. If the probability of the first, second, third and the fourth shot hitting the enemy aircraft is \[0.7,0.6,0.5\] and \[0.4\], what is the probability that four shots aimed at enemy aircraft will bring aircraft down?
1.\[0.084\]
2.\[0.916\]
3.\[0.036\]
4.\[0.964\]

Answer 1

Hint: In the given question, we have been given that there is a gun which fires four shots, and the probabilities of the shots hitting the enemy aircraft have been given – \[0.7,0.6,0.5\] and \[0.4\]. Now, it has been given that the gun is fired and as presumed, four shots propel out of it. Here, we have to calculate that when these four shots are fired, what is the probability that four shots will bring it down. Now, even if one shot hits the aircraft, it is going down. So, we have to calculate the probability that at least one shot hits the aircraft. This problem is most effectively solved using the complement of an event.

Formula Used:
Let the probability of an event \[E\] to occur be \[P\], then the probability of the event that \[E\] will not occur, more commonly known as that the complement of the event \[E\], is given by
\[P'\left( E \right) = 1 - P\]

Complete step-by-step answer:
In this question we have to calculate the probability that at least one shot is going to hit the aircraft. Now, this question can be solved by individually solving each case which fulfils the given condition – shot one hits only, shots one and two hit only, and so on. But this is a very lengthy method.
Instead of that, we can use the complement of a probability to evaluate the result much faster.
We have to calculate the probability that at least one shot hits the aircraft. And the complement (inverse) of this is that no shot hits it.
We are going to approach it by first calculating the probability that no shot hits the aircraft, which is quite easy to do. For that, we need the complement of the individual probabilities (\[0.7,0.6,0.5\] and \[0.4\]) and they are going to be – \[\left( {0.3,0.4,0.5,0.6} \right)\]. This complement probability is going to be,
\[P\left( {no\,{\rm{ }}\,shot{\rm{ }}\,hits{\rm{ }}\,the{\rm{ }}\,aircraft} \right) = 0.3 \times 0.4 \times 0.5 \times 0.6 = 0.036\]
Now, our required probability is the inverse of this probability and is,
\[P\left( {at\,{\rm{ }}\,least{\rm{ }}\,one{\rm{ }}\,shot{\rm{ }}\,hits} \right) = 1 - P\left( {no\,{\rm{ }}\,shot{\rm{ }}\,hits{\rm{ }}\,the{\rm{ }}\,aircraft} \right) = 1 - 0.036 = 0.964\]
Hence, the correct option is (4).



Note: So, for solving questions of such type, we first write what has been given to us. Then we write down what we have to find. Then we think about the formulae which contains the known and the unknown and pick the one which is the most suitable for the answer. Then we put in the knowns into the formula, evaluate the answer and find the unknown. It is really important to follow all the steps of the formula to solve the given expression very carefully and in the correct order, because even a slightest error is going to make the whole expression awry and is going to give us an incorrect answer.