Probability at least one die shows a six

Question

You roll two fair six-sided dice. What is the probability that at least one die shows a 6?

Accepted Answer

The tempting wrong answer is 	frac16 + 	frac16 = 	frac26 = 	frac13, which double-counts the outcome where both dice are 6. Count on the 36-cell grid of ordered outcomes instead. The cleanest route is the complement: the probability that neither die is a 6 is P(	ext{no six}) = \frac{5}{6} 	imes \frac{5}{6} = \frac{25}{36}, since each die independently avoids 6 with probability 5/6. Therefore P(	ext{at least one six}) = 1 - \frac{25}{36} = \frac{11}{36} \approx 0.306. Direct check: the outcomes with a 6 are the six in row "first die =6" plus the six in column "second die =6," minus the o…

Probability at least one die shows a six

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