Fast(er) way of computing the cumulative binomial probability?While revising for my probability test, I saw this question from one of the previous exams:You flip a fair coin 100 times. What is the probability that you have less than 45 heads?My question is a simple one. I know that to calculate the probability of a specific amount of n heads (or tails), we can use the binomial distribution with formula P ( X = k ) = ( n k ) p k ( 1 − p ) n − k , with n = 100 and p = 0.5 in this case. I also know that we can compute the chance of P ( X &lt; k ) as either P ( X = 0 ) + P ( X = 1 ) + . . . + P ( X = k − 10 ) or 1 − ( P ( X = k ) + P ( X = k + 1 ) + . . . + P ( X = n ) )However this is a simple question only worth two points out of over 60 total points. I cannot imagine that you need to compute P ( X = n ) 44 separate times and sum them together for such a small amount of points.Is there any way to rewrite the formula or apply a different trick to drastically lower the amount of computations you need to do?

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Fast(er) way of computing the cumulative binomial probability?While revising for my probability test, I saw this question from one of the previous exams:You flip a fair coin 100 times. What is the probability that you have less than 45 heads?My question is a simple one. I know that to calculate the probability of a specific amount of n heads (or tails), we can use the binomial distribution with formula   P  (  X  =  k  )  =                              (                            n        k                              )                          p    k    (  1  −  p      )          n      −      k      , with   n  =  100 and   p  =  0.5 in this case. I also know that we can compute the chance of   P  (  X  &amp;lt;  k  ) as either   P  (  X  =  0  )  +  P  (  X  =  1  )  +  .  .  .  +  P  (  X  =  k  −  10  ) or   1  −  (  P  (  X  =  k  )  +  P  (  X  =  k  +  1  )  +  .  .  .  +  P  (  X  =  n  )  )However this is a simple question only worth two points out of over 60 total points. I cannot imagine that you need to compute   P  (  X  =  n  ) 44 separate times and sum them together for such a small amount of points.Is there any way to rewrite the formula or apply a different trick to drastically lower the amount of computations you need to do?

Denier5h · Accepted Answer

Step 1The trick is to indeed use the central limit theorem since we already have a large n.Since our distribution is a binomial one, we have   E  [  X  ]  =  μ  =  0.5  ⋅  100  =  50 and   V  a  r  (  X  )  =      σ    2    =  0.5  ⋅  100  (  1  −  0.5  )  =  25Step 2Then, we can compute             45      −      μ                      σ        2              =            45      −      50              25        =  −  1. Looking this up in the Z-table gives 0.1587. This is close to the exact answer of 0.14 (and indeed, both 0.14 and 0.16 are marked as correct in the answer sheet).

Fast(er) way of computing the cumulative binomial probability? You flip a fair coin 100 times. What is the probability that you have less than 45 heads?

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