Poisson paradigm I have some problems with this
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.
Could you help me to understand?
 A: Let $ X_i $ be a Bernoulli random variable that represents the outcome of the $ i^{th} $ antiballistic missile, i.e. $ X_i = 1 $ if the antiballistic missile hits any one of the targets and $ X_i = 0 $ if the antiballistic missile misses all of the targets.
Since there are 10 targets, then the probability of hitting a given target (i.e. missile 1, 2, ... , or 10) is equal to: $ p = \frac{0.1}{10} = 0.01 $.
Let $ Y_i $ be the total number of missiles that hit the $ i^{th} $ target. According to the Poisson paradigm, the distribution of $ Y_i$ can be approximated by a Poisson distribution with parameter $ \lambda = E[Y_i] = E[\sum_{i=1}^n X_i] = \sum_{i=1}^n E[X_i] = nE[X] = 500 \times 0.01 = 5 $.
Next, we want to compute the probability that the $ i^{th} $ target is hit. This occurs when $ Y_i \geq 1 $, i.e. 
$$ P(Y_i \geq 1) = 1 - P(Y<1) = 1 - P(Y_i=0) = 1 - \frac{e^{-5}5^0}{0!} = 1 - e^{-5} $$
Finally, the probability that all 10 targets are hit is given by:
$$ P = \prod_{i=1}^{10}P(Y_i \geq 1) = \prod_{i=1}^{10}(1-e^{-5}) = (1-e^{-5})^{10} \approx 0.9346 $$
