For the cold-vaccination study in Example 9.1 (see Table...

For the cold-vaccination study in Example 9.1 (see Table 9.1), if $S=\{$ the 160 people \}, $A=\{$ got a cold\}, $A^{\prime}=$ \{did not get a cold \}, $B=\{$ was vaccinated $\}, B^{\prime}=$ \{was not vaccinated $\}$, then use equation (9.10) (the general addition rule) to determine: (a) $P(A \cup B)$, (b) $P\left(A^r \cup B\right)$, (c) $P\left(A \cup B^{\prime}\right)$, (d) $P\left(A^{\prime} \cup B^{\prime}\right)$, (e) $P\left(A \cup A^{\prime}\right)$

For the cold-vaccination study in Example 9.1 (see Table 9.1), if $S=\{$ the 160 people \}, $A=\{$ got a cold\}, $A^{\prime}=$ \{did not get a cold \}, $B=\{$ was vaccinated $\}, B^{\prime}=$ \{was not vaccinated $\}$, then use equation (9.10) (the general addition rule) to determine:
(a) $P(A \cup B)$,
(b) $P\left(A^r \cup B\right)$,
(c) $P\left(A \cup B^{\prime}\right)$,
(d) $P\left(A^{\prime} \cup B^{\prime}\right)$,
(e) $P\left(A \cup A^{\prime}\right)$

Key Concepts

Probability and Sample Space

Probability is a measure of how likely an event is to occur, defined over a sample space which comprises all possible outcomes. A clear understanding of the sample space is essential for accurately assigning and calculating probabilities associated with events, particularly when using set operations to describe these events.

Event Complements

The concept of an event complement is essential in probability theory, representing all outcomes that do not belong to a given event. Calculating the probability of an event's complement is often simpler and is computed as one minus the probability of the event itself, simplifying many probability calculations.

General Addition Rule

The general addition rule is a fundamental principle in probability that allows one to calculate the probability of the union of two events. It states that the probability of either event occurring is equal to the sum of their individual probabilities minus the probability of both events occurring simultaneously. This concept is critical for avoiding the double-counting of outcomes that occur in both events.

Set Operations in Probability

Set operations such as union, intersection, and complement form the foundation of probability theory by providing a structured way to describe and combine events. The union of two events includes any outcomes that are in either event, the intersection includes outcomes common to both, and the complement includes all outcomes not in the specified event.

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