The Sample Space of a Repeated Experiment: Understanding Mutually Exclusive Events

When we think of an experiment, we often think of a single event with a specific outcome. But what happens when we repeat an experiment until a certain event occurs? In this blog post, we're going to explore the sample space of a repeated experiment where events E and F are mutually exclusive.

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)].

First, let's define what we mean by mutually exclusive events. In a sample space, mutually exclusive events are events that cannot happen at the same time. For example, if we're flipping a coin, the event E could be "heads" and the event F could be "tails." These events are mutually exclusive because you can't get both heads and tails at the same time.
Now, let's imagine that we're repeating this coin flip experiment until either heads or tails comes up. In this case, the sample space of the new "super experiment" would be the set of all possible sequences of coin flips that end in either heads or tails. For example, one possible sequence could be "heads, tails, heads, heads" and another could be "tails, tails, heads."

So, what's the probability of event E (heads) occurring before event F (tails)? To calculate this, we can use the formula P(E) / [P(E) + P(F)]. In the case of our coin flip experiment, P(E) is 0.5 (the probability of getting heads on one flip) and P(F) is also 0.5 (the probability of getting tails on one flip). So, the probability of getting heads before tails is 0.5 / (0.5 + 0.5) = 0.5.

In conclusion, when we repeat an experiment until a certain event occurs, the sample space of the new super experiment is the set of all possible sequences of events, and the probability of one event occurring before another mutually exclusive event is P(E) / [P(E) + P(F)]. Understanding the sample space of a repeated experiment can help us better understand the probabilities of different outcomes and make more informed decisions.