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Situation:
   2000 skiers,
   each with probability .002 of an accident.
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# Exact Binomial probability for P(X = 2)

> dbinom(2,2000,.002)
[1] 0.1464517

# Poisson approximation for P(X = 2)

> dpois(2,4)
[1] 0.1465251

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# Exact Binomial probability for P(X <= 2)

> pbinom(2,2000,.002)
[1] 0.23781

# Poisson approximation for P(X <= 2)

> ppois(2,4)
[1] 0.2381033


--------------------------------------------------------
Modified Situation:
        2000 skiers with varying accident probabilities
        p[i] = 2e-6*i
---------------------------------------------------------

exact = exact answer computed by convolution
sim = simulation approximation based on 10^7 repetitions
poisson = general Poisson approximation.

           exact       sim      poisson
  0 1.818155e-02 0.0181944 1.827904e-02
  1 7.295722e-02 0.0730285 7.315273e-02
  2 1.462803e-01 0.1461901 1.463786e-01
  3 1.953991e-01 0.1953864 1.952691e-01
  4 1.956277e-01 0.1956419 1.953667e-01
  5 1.565806e-01 0.1565636 1.563715e-01
  6 1.043696e-01 0.1044772 1.042998e-01
  7 5.958988e-02 0.0594897 5.962969e-02
  8 2.975012e-02 0.0297902 2.982975e-02
  9 1.319355e-02 0.0131523 1.326430e-02
 10 5.262427e-03 0.0052818 5.308371e-03
 11 1.906895e-03 0.0019062 1.931282e-03
 12 6.329767e-04 0.0006253 6.440825e-04
 13 1.938185e-04 0.0001964 1.982783e-04
 14 5.507146e-05 0.0000536 5.667928e-05
 15 1.459496e-05 0.0000158 1.512203e-05
 16 3.623756e-06 0.0000054 3.782398e-06
 17 8.462410e-07 0.0000011 8.904210e-07
 18 1.865149e-07 0.0000000 1.979703e-07
 19 3.891881e-08 0.0000001 4.169879e-08
 20 7.709689e-09 0.0000000 8.343928e-09