4.3

$$P(B \mid A)$$

The probability of event $B$ occurring, given that event $A$ has already occurred.

Complement rule formula

P(not A) = 1-P(A)

False negative

A test result that indicates a condition is absent when it is actually present.

Formula for P(at least one success) in $n$ independent trials with success probability $p$

P(at least one) = 1 - (1-p)^n

Prior probability

An initial probability value evaluated before receiving any new or additional information.

Vertical bar symbol ($\mid$) in probability

Represents the phrase "given that", denoting a conditional event.

Bayes' theorem formula

$$P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}$$

Meaning of a probability equal to $0$

The event is impossible and cannot occur.

Prevalence

The proportion or percentage of a population that possesses a specific condition.

Method to find $P(A \mid B)$ from a contingency table

Divide the count of outcomes where both $A$ and $B$ occur by the total count for the given condition $B$.

Mathematical meaning of the phrase "at least one"

One or more occurrences ($\ge 1$).

Confusion of the inverse

The erroneous assumption that $P(A \mid B) = P(B \mid A)$

True positive

A test result that correctly indicates a condition is present when it is actually present.

Formula for conditional probability $P(B \mid A)$ in terms of joint probability

$$P(B \mid A) = \frac{P(A \text{ and } B)}{P(A)}$$

Posterior probability

A probability value that is revised or updated after taking into account new information.

$$P(A \text{ and } B)$$

The joint probability that both event $A$ and event $B$ occur simultaneously.

Formula for the probability of zero occurrences in $n$ independent trials

$$P(\text{none}) = (1 - p)^n$$

Primary application of Bayes' theorem

To calculate a revised (posterior) probability by incorporating new information or evidence.

Conditional probability

A probability calculated under the condition or constraint that another event has already occurred.

Complement of the event "at least one occurrence"

Zero occurrences (no occurrences).

Meaning of a probability equal to $1$

The event is certain to occur.

Role of $P(A)$ in the conditional expression $P(B \mid A)$

The probability of the given condition or event ($A$).

True negative

A test result that correctly indicates a condition is absent when it is truly absent.

Advantage of using the complement rule for "at least one"

Finding the probability of zero occurrences ($P(\text{none})$) is often much simpler than calculating and summing the probabilities of one or more occurrences.

General relationship between $P(A \mid B)$ and $P(B \mid A)$

Conditional probabilities are generally not equal ($P(A \mid B) \neq P(B \mid A)$

General complement formula for "at least one"

$$P(\text{at least one}) = 1 - P(\text{none})$$

False positive

A test result that incorrectly indicates a condition is present when it is actually absent.

Difference between prior probability and posterior probability

A prior probability is an initial assessment before new evidence is known, while a posterior probability is the updated assessment after considering new evidence.

Meaning of $P(B)$ in Bayes' theorem $P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}$

The overall or marginal probability of event $B$ occurring across all conditions.

Intuitive assumption when finding $P(B \mid A)$

Assume event $A$ has already occurred, restricting the sample space to only outcomes in $A$.

Procedure for finding conditional probability using a two-way table

Restrict the focus strictly to the row or column of the given event, then divide the target intersection count by that row or column total.

Verbal interpretation of $P(B \mid A)$

The likelihood of event $B$ occurring after knowing that event $A$ has occurred.

Complement of an event $A$

All outcomes in the sample space in which event $A$ does not occur.

Core concept of conditional probability

Adjusting or calculating a probability based on specific information about another event.

Ratio equal to $P(B \mid A)$ using joint probability

$$\frac{P(A \text{ and } B)}{P(A)}$$