How many rolls to be 90% sure of a six?

Question

You roll a fair die repeatedly. How many rolls do you need so that the probability of at least one six is at least 90\%?

Accepted Answer

With n independent rolls, P(	ext{at least one six}) = 1 - (5/6)^n. We want 1 - \left(	frac{5}{6}ight)^n \ge 0.9 \quad\Longleftrightarrow\quad \left(	frac{5}{6}ight)^n \le 0.1. Take logs (both sides, noting \ln(5/6) < 0 flips the inequality): n \ge \frac{\ln 0.1}{\ln (5/6)} = \frac{-2.303}{-0.182} \approx 12.6. Since n must be a whole number, round up to n = 13. Checking: (5/6)^{13} \approx 0.093, giving P \approx 0.907 \ge 0.9, while (5/6)^{12} \approx 0.112 gives only P \approx 0.888. So 13 rolls just clears the bar. Takeaway. Inverting an "at least one" question is a one-line log sol…

How many rolls to be 90% sure of a six?

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