Probabilities of sequences of independent events

A sequence of independent attempts is observed

A success is an attempt with the required result. Let \(E_i\) denote success on trial \(i\)

The trial outcomes are independent, and the sample space is \(S\)

The exact success probabilities are \(\Pr(E_i) = \frac{1}{3} \,\text{ for every trial}\)

Let \(A\) be the event that no trial is a success