What is the probability of rolling a 6 on a fair six-sided die?

Step-by-step solution

1. 1

   List the sample space. A single roll of a six-sided die has outcomes

   $S=\{1,2,3,4,5,6\},\qquad |S|=6$

   The word fair is what licenses the next step: it means all six outcomes are equally likely, each with probability $\tfrac16$.

2. 2

   Identify the favourable outcomes. The event "roll a $6$" contains exactly one outcome:

   $E=\{6\},\qquad |E|=1$

3. 3

   Apply the equally-likely-outcomes formula.

   $P(E)=\frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}=\frac{1}{6}$

4. 4

   Express the answer in decimal and percent form.

   $\frac16=0.1\overline{6}\approx 0.1667\approx 16.67\%$

5. 5

   Check the probabilities sum to 1, and note what the answer does not mean. Each of the six faces has probability $\tfrac16$, and $6\times\tfrac16=1\;\checkmark$. The value $\tfrac16$ is a long-run frequency, not a guarantee: rolling six times gives at least one $6$ with probability $1-\left(\tfrac56\right)^6\approx 0.665$, not $1$.