KA-0168Complementary counting
5 points, difficulty 3 of 3Level 9–10about 150sAn ordinary six-sided die is rolled three times. What is the probability that at least one roll shows a six?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
At least one is much easier to count by asking about the opposite.
2The strategy
The opposite of at least one six is no sixes at all in three rolls.
3The full solution
Each roll avoids a six with probability 5/6, so no sixes has probability (5/6)^3 = 125/216, and the answer is 1 - 125/216 = 91/216.
Solution
The reliable way
The phrase at least one is a signal to count the complement. The opposite event is no six on any of the three rolls. Each roll avoids a six with probability 5/6, and the rolls are independent, so no sixes has probability (5/6)^3 = 125/216. The event we want is everything else: 1 - 125/216 = 91/216, which is a little over 42 percent. The transferable idea: at least one almost always means one minus none.
The elegant way
One minus (5/6)^3 = 1 - 125/216 = 91/216.
Why this is on the test: Adding 1/6 three times is the classic error, and it visibly breaks at seven rolls where it would give a probability above 1.