A coin is to be tossed as many times as necessary to turn up...

A coin is to be tossed as many times as necessary to turn up one head. Thus the elements $c$ of the sample space $\mathcal{C}$ are $H, T H, T T H$, TTTH, and so forth. Let the probability set function $P$ assign to these elements the respective probabilities $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}$, and so forth. Show that $P(\mathcal{C})=1 .$ Let $C_{1}=\{c:$ c is $H, T H, T T H$, TTTH, or TTTTH $\}$. Compute $P\left(C_{1}\right)$. Next, suppose that $C_{2}=$ $\{c: c$ is TTTTH or TTTTTH $\}$. Compute $P\left(C_{2}\right), P\left(C_{1} \cap C_{2}\right)$, and $P\left(C_{1} \cup C_{2}\right)$.

Key Concepts

Sample Space

In probability theory, the sample space is the set of all possible outcomes of an experiment. In this case, the sample space is defined by the sequences of coin tosses required to produce the first head. Understanding how the sample space is constructed is fundamental to determining the probability of events, especially when the sample space is countably infinite as in this example.

Probability Measure

A probability measure is a function that assigns a probability to each event in a sample space, ensuring that the total probability sums to one. In this example, each outcome given by successive coin tosses is assigned a probability according to a rule (specifically a geometric progression), illustrating how the measure is used to model real world or theoretical experiments.

Geometric Series in Probability

The use of a geometric series arises when probabilities decrease exponentially with the number of events, as with repeated independent coin tosses. Recognizing and summing a geometric series allows us to show that the total probability of the entire sample space equals one, and to calculate the probabilities of specific events that are defined by subsets of the sample space.

Events, Unions, and Intersections

Events are subsets of the sample space, and operations such as unions and intersections are used to combine or identify overlapping events. The union of events represents outcomes that belong to either event, while the intersection represents outcomes common to both. This concept is key when computing compound probabilities and applying the additive rules of probability.

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