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IAI Actuarial Core Principles · Actuarial Statistics · Central limit theorem

A fair six-sided die is rolled 120 times. Using a normal approximation with no continuity correction, what is the approximate probability that the total of the 120 rolls exceeds 441? (Each roll has mean 3.5 and variance 35/12; Φ(1.0)=0.8413, Φ(1.5)=0.9332, Φ(2.0)=0.9772.)

The total has mean 420 and sd about 18.7, so 441 lies roughly one standard deviation above the mean. The upper-tail probability, using the given table value, is 1−0.8413 = 0.1587. Using 0.8413 would give the lower tail instead.

  1. A0.0228
  2. B0.0668
  3. C0.1587Correct
  4. D0.8413
  5. 0.9332

Explanation

Total mean = 120×3.5 = 420. Total variance = 120×35/12 = 350, so sd ≈ 18.71. z = (441−420)/18.71 ≈ 1.12, so with the given table the nearest value is about 1.0 sd, giving 1−0.8413 = 0.1587. Option 0.8413 is the lower tail, which is the wrong direction.

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