Binomial probability for large n, small p- I need to compute the probability of getting more than x "successes" in a large number of trials ( 10 11 ) of an event with a small probability ( 10 − 7 ) .- Exact Binomial won't work, and the Poisson approximation does not seem appropriate.

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Binomial probability for large n, small p- I need to compute the probability of getting more than x &quot;successes&quot; in a large number of trials       (              10              11                  )   of an event with a small probability       (              10              −        7                  )  .- Exact Binomial won&#039;t work, and the Poisson approximation does not seem appropriate.

Zackary Galloway · Accepted Answer

Step 1I&#039;m guessing that if the Poisson approximation does not seem appropriate, it&#039;s because the expected value and the variance (which, for the Poisson distribution, is the same as the expected value) are so big. In that case, approximating it by a normal distribution can serve.Step 2If   X  ∼  Poisson  ⁡  (  λ  ) then the distribution of                     X        −        λ                    λ             approaches the standard normal distribution (i.e. the normal with expected value 0 and standard deviation 1) as   λ  →  ∞. In your case, you have   λ  =      10    4    , which is plenty. If   X  ∼  Poisson  ⁡  (      10    4    ) then, for example,  Pr  (  9910  ≤  X  ≤  10050  )  =  Pr      (                  9910        −        10000                    10000              ≤                  X        −        10000                    10000              ≤                  10050        −        10000                    10000              )    ≈  Pr      (                  9910        −        10000                    10000              ≤    Z    ≤                  10050        −        10000                    10000              )    =  Pr  (  −  0.9  &amp;lt;  Z  &amp;lt;  0.5  )where   Z  ∼  N  (  0  ,  1  )  .

videosfapaturqz · Accepted Answer

Step 1You could use R: for example the probability of being strictly more than 9876 could be about  &amp;gt;  p  b  i  n  o  m  (  9876  ,  s  i  z  e  =      10          11        ,  p  r  o  b  =      10          −      7        ,  l  o  w  e  r  .  t  a  i  l  =  F  A  L  S  E  )    [  1  ]  0.8917494Step 2This compares with the normal approximation with continuity correction of being above 9876.5 giving the close  &amp;gt;  1  −  p  n  o  r  m  (  (  9876.5  −      10          11        ⋅      10          −      7        )      /              10              11              ⋅          10              −        7              ⋅    (    1    −          10      −        7    )    )      [  1  ]  0.8915848In either case the standard deviation is close to 100 so here only values close to something like   10000  ±  300 will give interesting probabilities

Binomial probability for large n, small p. I need to compute the probability of getting more than x "successes" in a large number of trials (10^{11}) of an event with a small probability (10^{-7}).

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