The best replacement for binomial distribution is poisson distribution.

The best alternatives to binomial distribution are rated as following:
★★★★★  poisson distribution
★★★★  normal distribution
★★★★  normal
★★★★  geometric
★★  probability distribution
★★  bernoulli
★  gaussian
★  negative binomial distribution
★  hypergeometric
★  poisson
★  geometric distribution
  nominal distribution
  sampling distribution

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binomial distribution
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normal distribution