You roll a die 1000 times, you add up the numbers you rolled, and you get 3689. Do you think it is a fair die? I knew that it should use the central limit theorem but I have no idea how to do that. Should I use a fair dice's expected values ie 3.5 and since each time roll a dice is independent and the mean is 3500 for a fair dice's expected value in 1000 times.But I have no idea what to do next
2026-03-30 08:35:57.1774859757
central limit theorem about dice
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The sum of the obtained numbers is a random variable with mean 3500(as you correctly found out) and it's standard deviation is approximately 54.006(I hope you can find it yourself).
Now the number you got, $3689$ is nearly $3.5$ times the standard deviation away from the mean. After looking up the table of error functions (assuming the number of trials $1000$ to be sufficiently high, this is where the Central Limit Theorem is used) we see that the probability that a Gaussian random variable lies in the interval $[\mu − 3\sigma, \mu + 3\sigma]$ is equal to $0.9973$. So the dice is (most likely) unfair.