In Chapter 8, we used the set theory interpretation of...

In Chapter 8, we used the set theory interpretation of probability to show for the experiment of flipping a coin twice that
$$
P(H H, H T, T H)=P(\text { at least one head })=0.75
$$

Now, use equation (9.8) (the special multiplication rule) to show that this is true.

Key Concepts

Complement Rule

The complement rule states that the probability of an event occurring is equal to one minus the probability of the event not occurring. It is especially useful for calculating probabilities of events described in terms like ‘at least one’ by subtracting the probability of the complementary event (e.g., none occurring) from one.

Independence of Events

In probability theory, independence means that the occurrence of one event does not affect the probability of another. This property is fundamental in many experiments and simplifies the calculation of joint probabilities, as it allows the special multiplication rule to reduce to a simple product of the probabilities of the individual events.

Special Multiplication Rule

The special multiplication rule is used to calculate the probability of the intersection of events, taking into account possible dependencies between events by using conditional probabilities. In cases where events are independent, this rule simplifies to the product of the probabilities of the individual events. It is crucial for sequential experiments where the outcome of one event may influence the subsequent one, or where the structure of the experiment naturally decomposes into a series of events.

Set Theory Interpretation of Probability

This concept frames outcomes as elements of a set (known as the sample space) and events as subsets of this sample space. It allows us to use set operations like unions, intersections, and complements to calculate probabilities, providing a systematic way to handle outcomes and events such as those emerging from multiple coin flips.

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