Consider again the experiment described in Example 9.1 in...

Consider again the experiment described in Example 9.1 in which a vaccine is tested on a group of 160 volunteers. Eighty volunteers are vaccinated and the rest are not. After 12 months all 160 people are asked if they got a cold during the past year. The results are shown in Table 9.1. The experiment is to randomly select one of the 160 people. If for this experiment $S=\{$ the 160 people\}, $A=\{$ got a cold $\}, A^{\prime}=\{$ did not get a cold $\}, B=\{$ vaccinated $\}$, and $B^{\prime}=\{$ not vaccinated $\}$, then find these probabilities: (a) $P\left(A \mid B^{\prime}\right)$, (b) $P(B \mid A)$, (c) $P\left(B^{\prime} \mid A^{\prime}\right)$.

Consider again the experiment described in Example 9.1 in which a vaccine is tested on a group of 160 volunteers. Eighty volunteers are vaccinated and the rest are not. After 12 months all 160 people are asked if they got a cold during the past year. The results are shown in Table 9.1. The experiment is to randomly select one of the 160 people. If for this experiment $S=\{$ the 160 people\}, $A=\{$ got a cold $\}, A^{\prime}=\{$ did not get a cold $\}, B=\{$ vaccinated $\}$, and $B^{\prime}=\{$ not vaccinated $\}$, then find these probabilities:
(a) $P\left(A \mid B^{\prime}\right)$,
(b) $P(B \mid A)$,
(c) $P\left(B^{\prime} \mid A^{\prime}\right)$.

Key Concepts

Contingency Tables and Data Organization

Contingency tables are a statistical tool used to organize and summarize data from experiments or surveys where observations are classified according to two or more categorical variables. They help in visualizing the frequency distribution of variables and are particularly useful for calculating probabilities, including conditional probabilities, by providing a clear breakdown of counts in each category. This method of organizing data facilitates a better understanding of relationships between variables and aids in hypothesis testing and data interpretation.

Sample Space and Events

The sample space is the set of all possible outcomes of a random experiment, and events are subsets of this sample space that we are interested in. Defining a clear sample space and events allows for accurate modeling and probability calculation, as each outcome must be accounted for. Understanding these foundational elements is critical in probability theory because they provide the structure upon which probability measures and further analyses, such as conditional probabilities, are built.

Conditional Probability

Conditional probability is the measure of the probability of an event occurring given that another event has already occurred. It is calculated by dividing the probability of the intersection of two events by the probability of the conditioning event, provided that the latter event has a nonzero probability. This concept is essential when analyzing situations where the occurrence of one event influences the likelihood of another, and it is widely applicable in areas such as decision making, risk assessment, and statistical inference.

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