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I am working on a binomial distribution problem, where I would like to calculate the probability of observing 48601000 successes in 69430000 trials, where the success probability is $p = 0.70$. I compute this using the binomial pmf: $$\binom{69430000}{48601000} 0.70^{48601000} (0.30)^{20829000} \approx 0.0001044785$$ Can anyone provide some intuition as to why this probability is so low, given that $p = 0.70$? Just trying to wrap my head around it!

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    $\begingroup$ It might be more revealing to invert your question: could you tell us why we should expect the probability of any particular number of successes to be so high, given that there are almost 70 million trials and therefore even more possibilities for the number of successes? $\endgroup$ – whuber Apr 19 '18 at 22:53
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    $\begingroup$ Hi. That is a good point. I suppose in some sense there is nothing "special" about a particular number of successes. $\endgroup$ – Thomas Moore Apr 19 '18 at 23:14
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Why are you surprised? The number of possible outcomes is very large, all with some positive probability, so the individual probabilities cannot be too large. We can calculate the maximum possible point probability for your specific binomial distribution, with some help from R:

n <- 69430000
p <- 0.7
max(dbinom(c((n+1)*p, (n+1)*p-1), n, p))
    0.0001044785

which is what you have given, because the number of successes you gave is the mode of that particular binomial distribution.

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