Question

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 $n$ th time is $P(E) \times(1-p)^{n-1}, n=1,2, \ldots$, where $p=P(E)+$ $P(F)$. Add these probabilities to get the desired answer.

          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 $n$ th time is $P(E) \times(1-p)^{n-1}, n=1,2, \ldots$, where $p=P(E)+$ $P(F)$. Add these probabilities to get the desired answer.

Added by Peter O.

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