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

LiftoffMachine Learning Engineer · Onsite
HardMachine Learning EngineerOnsiteStatistics & Math

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A fair six-sided die is rolled 1,000 times; estimate the probability that the sum of the faces exceeds a threshold, including a threshold of 600. Tests the mean and variance of one roll, how they scale for a sum of independent rolls, the normal approximation with a continuity correction, and range checks.

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 $X$.

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?

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 $t = 600$.

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

Hints

Hint 1

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.

Hint 2

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.

Hint 3

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 4

Look before you standardize: Before computing a z-score, ask whether the threshold is a plausible value for $S$ at all.

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?