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In mathematica, RandomInteger[] generates binary numbers (0 or 1) with equal probablity (uniform distribution).

If I wish to generate the binary numbers with unequal probability or from some probability distribution, let's say Binomial distribution, then how to generate them?

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For example, for 10 binomial numbers of length 5, you could use either BinomialDistribution (as shown by @Kuba in the comments) or RandomChoice

p = 1/10;

SeedRandom[0]

RandomVariate[BinomialDistribution[1, p], {10, 5}]

(* {{0, 0, 0, 0, 1}, {1, 0, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0,
   0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {1, 0, 0, 1, 0}, {1, 0, 0, 0, 
  0}, {0, 0, 0, 0, 0}} *)

SeedRandom[0]

RandomChoice[{1 - p, p} -> {0, 1}, {10, 5}]

(* {{0, 0, 0, 0, 1}, {1, 0, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0,
   0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {1, 0, 0, 1, 0}, {1, 0, 0, 0, 
  0}, {0, 0, 0, 0, 0}} *)

The approaches are equivalent

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