12. Let E and F be mutually exclusive events in the sample space of an experiment.

Suppose that the experiment is repeated until either event E or event F occurs.

What does the sample space of this new super experiment look like? Show that the probability that event E occurs before event F is P(E)/ [P(E) + P(F)].

Hint: Argue that the probability that the original experiment is performed n times and E appears on the nth time is P(E) × (1 − p)n−1, n = 1, 2,..., where p = P(E) + P(F). Add these probabilities to get the desired answer.