Q.4.68 - A First Course in Probability

Question

In response to an attack of $10$ missiles, $500$ antiballistic missiles are launched. The missile targets of the antiballistic missiles are independent, and each antiballstic missile is equally likely to go towards any of the target missiles. If each antiballistic missile independently hits its target with probability .$1,$ use the Poisson paradigm to approximate the probability that all missiles are hit. 

Step-by-Step Solution

Answer

The required probability is $0.93462.$

Step 1: Given information

Given in the question that, in response to an attack of $10$ missiles, $500$ antiballistic missiles are launched. The missile targets of the antiballistic missiles are independent, and each antiballistic missile is equally likely to go towards any of the target missiles. If each antiballistic missile independently hits its target with probability .$1,$ 

We need to use the Poisson paradigm to approximate the probability that all missiles are hit. 

Step 2: Explanation

Selected some missile out of ten of them and consider the probability that some antiballistic missile hits it. First of all, probability that some antiballistic missile chooses that fixed missile as its target is $0.1$since it chooses randomly.

Also, if that was the case, the probability that it has been hit by that antiballistic missile is $0.1$ (we were given that).

Thus, the probability that some antiballistic missile will hit our missile is $0.01$.

Define $X$ as the random variable which marks the number of antiballistic missile that have hit our fixed missile.

We use Poisson approximation to obtain that

$X~Pois(500·0.01)$

so the probability that our missile has been hit is simply

$P(X\ge 1)=1-P(X=0)=1-e^{-5}$

Because of the independence, the probability that all of the missiles have been hit is

$\left(P\right(X\ge 1){)}^{10}={\left(1-e^{-5}\right)}^{10}=0.93462$

Step 3: Final answer

Use Poisson approximation to obtain that the required probability is $0.93462.$