I know one can use the binomial distribution to calculate the likelihood of getting 0, 1, 2, etc heads and by summing these up to 30, the likelihood of getting less than 30 heads out of 100. But can you specifically use Central Limit Theorem to find the same likelihood?

  • $\begingroup$ Strictly speaking you cannot, simply because the CLT is a theorem about a limit. You can, however, use the Normal approximation to the Binomial distribution. $\endgroup$
    – whuber
    Apr 3, 2018 at 13:51
  • $\begingroup$ Does one need the CLT to show that the Binomial Distribution is approximated as the Normal Distribution. $\endgroup$ Apr 3, 2018 at 14:38
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    $\begingroup$ No. Often the CLT is invoked as intuitive motivation for the approximation, but it provides no mathematical rigor. Approximations are useless anyway unless they are accompanied by quantitative statements of how good they are in any particular circumstance. The usual formulations of the CLT do not do that: they merely assert something happens in the limit. $\endgroup$
    – whuber
    Apr 3, 2018 at 14:58
  • $\begingroup$ Are you assuming that the probability of a head on any individual toss is 1/2? $\endgroup$ Apr 3, 2018 at 16:33


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