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I am reading through a book teaching Binomial Probability. Using the Binomial Distribution Formula , they define ‘$n$’ as being $n$ independent trials of the same experiment. Which makes sense. Rolling the same dice $6$ times is $n = 6$. However I got confused when the book presented me with this problem: “six dice are rolled. Find the probability that two of them show a four” answer is $0.200939$, but only if one considers rolling $6$ dice at the same time as $6$ trials of the same experiment. In my mind throwing six dice at the same time is a specific experiment done only one time, thus $n = 1$. Maybe I am misinterpreting the question. What am I missing in my understanding?

HeroZhang001
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ille
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  • Yes, it is assumed that each die result is independent, with probability of $\frac16$ for showing each of the $6$ faces, so it amounts to the same as throwing one die $6$ times. – true blue anil Feb 12 '23 at 08:33
  • Side note: The formulation flip a coin n times is better than flip n coins one, as the former allows one to consider things like the ballot theorem. – Andrew Feb 12 '23 at 09:07
  • Give this answer (in particular, its final paragraph) a read: An experiment's events don't have an inherent sequence. – ryang Feb 12 '23 at 13:18
  • You can define $n$ each time you use it. If a binomial probability here is ${n \choose k}p^k(1-p)^{n-k}$ with $n=6$, $k=2$, $p=\frac16$ it does not matter whether you threw one fair die six times or six fair dice one time each – Henry Feb 12 '23 at 14:21

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The Binomial distribution is defined as the number of successes in $n$ independent Bernoulli trials (experiments). So $n$ is number of Bernoulli trials. Nothing is assumed about organizing Bernoulli trials, they need to be independent and have fixed probability of success only.

kludg
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