Dice: Inclusion-Exclusion Principle
A fair six-sided die is rolled 6 times. What is the probability of 5 and 6 appearing at least once?
Answer
This probability is 1 minus the probability of a 5 not appearing OR a 6 not appearing. Using the inclusion-exclusion principle, the probability is:
\[
1 - 2\left(\frac{5}{6}\right)^6 + \left(\frac{4}{6}\right)^6.
\]
Numerically, this is:
\[
1 - 2\left(\frac{15625}{46656}\right) + \frac{4096}{46656} = \frac{19502}{46656} \approx 0.418.
\]