Probabilities of sequences of independent events
A synthetic sample contains independent binary observations
A success is the designated observation. Let \(E_i\) denote success on trial \(i\)
The trial outcomes are independent, and the sample space is \(S\)
The exact success probabilities are \(\Pr(E_i) = \frac{2}{3} \,\text{ for every trial}\)
Let \(A\) be the event that every trial is a success
\[A = E_{1}\cap E_{2}\cap E_{3}\]