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In response to an attack of 10 missiles, a response of 500 antiballistic missiles are launched. The missile targets of each antimissile are independent, and equiprobably picked. Each antimissile has a $.1$ chance to hit. What is the probability of all 10 missiles being hit?

The question then asks the student to use a Poisson process, although I'm willing to solve this using other methods. I thought I'd have

$$X=\sum X_i$$

with

$$X_i=1,0$$

if hit or not hit respectively. Then

$$P(X_i=1)=\sum_{k=1}^{500}\,_{500}\text{C}_kp^k(1-p)^{500-k}$$

and $p=1/100$.

I'm fairly certain my approach is wrong, and I do not know how to finish the problem.

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  • $\begingroup$ I have seen this question before on this site. I don't have the link at the moment. $\endgroup$ – Chinny84 Mar 11 '15 at 23:49
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Exact Answer

For the exact answer, I would approach this using Inclusion-Exclusion.

Let $N_j$ be the event that incoming missile $j$ is missed.

The intersection of $k$ of the $N_j$ is $k$ particular missiles being missed. There is a $\frac{k}{10}$ chance that any will be targeted by a given anti-missile and a $\frac1{10}$ chance that that anti-missile will hit. Thus, there is a $\left(1-\frac{k}{10}\cdot\frac1{10}\right)$ chance that that anti-missile will miss, and a $\left(1-\frac{k}{100}\right)^{500}$ chance that all $500$ will miss. There are $\binom{10}{k}$ ways to choose the $k$ particular missiles Thus, the sum of the probabilities of $k$ missiles being missed by $500$ anti-missiles is $$ \binom{10}{k}\left(1-\frac{k}{100}\right)^{500} $$ Inclusion-Exclusion says that the probability of some missile not being hit is $$ \sum_{k=1}^{10}(-1)^{k-1}\binom{10}{k}\left(1-\frac{k}{100}\right)^{500}=0.0638876 $$ Thus, the probability of all $10$ missiles being hit is $0.9361124$


Poisson Approximation

Since there are $500$ anti-missiles, each with a probability of $\frac1{100}$ of targeting a particular incoming missile, the expected number of hits on each missile is $5$. Thus, the Poisson approximation for getting no hits on a particular incoming missile is $e^{-5}$. Therefore, the probability of getting at least one hit on each particular missile is $\left(1-e^{-5}\right)$ and the probability of hitting all $10$ is $$ \left(1-e^{-5}\right)^{10}=0.9346272 $$ This is pretty close to the exact answer.

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  • $\begingroup$ These answers makes sense to me. I appreciate the thoroughness. Thanks! $\endgroup$ – ok_ Mar 12 '15 at 2:01
  • $\begingroup$ @robjohn Could one also say that the probability a particular missile is hit is $(1-(1-0.01)^{500})^{10}=0.9362042...$? Or is this wrong? It seems to be really close to the answer, is this just a fluke? $\endgroup$ – Guacho Perez Dec 8 '17 at 2:55

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