Combinatorial Sample Spaces

We can build complex sample spaces out of simpler ones.

• Let sample space S₁ be the outcomes of a single coin-flip experiment: $S_1 = \{ \mbox{heads}, \mbox{tails} \}$, with $\Pr\{\mbox{heads}\} = \Pr\{\mbox{tails}\} = 1/2$ (uniform distribution).
• Let S₂ be the outcomes of an ordered pair of coin-flip experiments: $S_2 = S_1 \times S_1 = \{ \mbox{heads heads}, \mbox{heads tails}, \mbox{tails heads}, \mbox{tails tails} \}$.
• Assuming independent and identically distributed (i.i.d.) trials, we can infer, e.g., that $\Pr\{\mbox{tails heads}\} = \Pr\{\mbox{tails}\} \Pr\{\mbox{heads}\} = 1/4$. The resulting distribution is again uniform.

Similarly, we can define a uniform distribution over the sample space of permutations out of a smaller number of simpler sample spaces.