Probability That the Sum of 1,000 Fair Die Rolls Exceeds a Threshold

A fair six-sided die is rolled 1,000 times, and the rolls are independent. Let $S$ be the sum of the 1,000 faces. Estimate the probability that $S$ exceeds a threshold $t$: give a general expression in $t$, and then the value for $t = 600$. ### Clarifying Questions - Does "exceeds" mean strictly greater than $t$, or greater than or equal to it? - Is an approximation acceptable, or is an exact value expected? - Is a normal table or calculator available, or should the final number be estimated by hand? ### Part 1 — One roll Let $X$ be the value shown by one roll. Find the mean, the variance and the standard deviation of $X$. ```hint Second moment first The variance follows from $\mathbb{E}[X^2]$ and $\mathbb{E}[X]$. Keep the variance and the standard deviation clearly apart; they are easy to swap here. ``` #### What This Part Should Cover - The mean, by direct summation or by symmetry - The variance via the second moment, as an exact fraction - The distinction between variance and standard deviation ### Part 2 — The sum of 1,000 rolls Find the mean, the variance and the standard deviation of $S$. Which distribution does $S$ approximately follow, and why? ```hint What grows with the number of rolls Decide which of the mean, the variance and the standard deviation grows in proportion to the number of rolls, and what independence contributes to that. ``` #### What This Part Should Cover - Linearity of expectation for the mean - Additivity of variance for independent rolls, and the resulting growth of the standard deviation - The limit theorem that justifies the approximate distribution, and its conditions ### Part 3 — The probability Write $P(S > t)$ in terms of $\Phi$, the CDF of the standard normal distribution. Account for the fact that $S$ takes only integer values. Then give the probability for $t = 600$. ```hint Integers on a continuous curve For an integer $t$, the event $S > t$ is the same as an event about the next integer. Think about where to place the cut on a continuous curve so that it represents that event fairly. ``` ```hint Look before you standardize Before computing a z-score, ask whether the threshold is a plausible value for $S$ at all. ``` #### What This Part Should Cover - Standardizing the threshold with the mean and standard deviation of $S$ - A continuity correction applied in the right direction - A check of $t = 600$ against the possible range of $S$ - When the approximation is reliable and when it is not ### What a Strong Answer Covers - Exact single-roll moments, with variance and standard deviation never confused - Correct scaling from one roll to 1,000 rolls - A clean expression for $P(S > t)$ through $\Phi$, including the continuity correction - A range check before any approximation is applied - Awareness of approximation error in the tails, and an exact method as a fallback ### Follow-up Questions - How does the answer for the threshold 600 change if the die is rolled only 100 times? - How accurate is the normal approximation for a threshold far above the mean, and what would you use there instead? - Without a table or calculator, how would you estimate the probability for a threshold about two standard deviations above the mean? - How would you compute $P(S > t)$ exactly for 1,000 rolls?