Statistics Chapter 15: General Addition Rule

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Questions and Answers

Which addition rule should be used when two events are not disjoint?

the general addition rule

What does the general addition rule state?

P(A or B) = P(A) + P(B) - P(A and B)

When we want the probability of an event from a conditional distribution, what do we write?

P(B|A)

How do we pronounce the term P(B|A)?

the probability of B given A Signup and view all the answers

What is a conditional probability?

a probability that takes into account a given condition Signup and view all the answers

What equation describes conditional probability?

P(B|A) = P(A and B) / P(A) Signup and view all the answers

What is one restriction encountered with the conditional probability formula?

the formula doesn't work if A is equal to zero Signup and view all the answers

Which multiplication rule should be used when two events are not independent?

the general multiplication rule Signup and view all the answers

What is the general multiplication rule?

P(A and B) = P(A) x P(B|A) Signup and view all the answers

What is our informal definition of independence?

the outcome of one event does not influence the probability of the other Signup and view all the answers

What is the formal definition of independence?

Events A and B are independent whenever: P(B|A) = P(B) Signup and view all the answers

Can disjoint events ever be independent?

False Signup and view all the answers

Why can't disjoint events be independent?

since we know that disjoint events have no outcomes in common, knowing that one occurred means the other didn't. Signup and view all the answers

What should you stop and do whenever you see probabilities multiplied together?

stop and ask whether you think that they are really independent Signup and view all the answers

When is it a good idea to use a tree diagram?

almost any time that you plan to make use of the General Multiplication Rule Signup and view all the answers

What is the sensitivity of a test?

the proportion of people who correctly test positive when they actually have the disease Signup and view all the answers

What is the specificity of a test?

the proportion of people who correctly test negative when they don't have the disease Signup and view all the answers

What is the base rate of a disease?

the probability that you are likely to have the disease, without any knowledge of your test results Signup and view all the answers

What is the positive predictive rate (PPR) of a test?

the probability of being diseased, given that someone tests positive Signup and view all the answers

Conceptually, what is Bayes' Rule?

a formula for reversing the probability from the conditional Signup and view all the answers

Study Notes

General Addition Rule

Applicable when two events are not disjoint.
Formula: P(A or B) = P(A) + P(B) - P(A and B).

Conditional Probability

Denoted as P(B|A), meaning the probability of event B given event A.
Reflects a probability that considers an existing condition.
Formula for calculating: P(B|A) = P(A and B) / P(A).
Formula is not applicable if P(A) is zero.

General Multiplication Rule

Used when two events are not independent.
Defined as: P(A and B) = P(A) × P(B|A).

Independence of Events

Informal definition: The result of one event does not affect the another’s probability.
Formal definition: Events A and B are independent if P(B|A) = P(B).
Disjoint events cannot be independent as their occurrence affects the probability of each other.
When probabilities are multiplied, verify the independence of the events.

Tree Diagrams

Useful for visualizing scenarios involving the General Multiplication Rule.

Test Sensitivity and Specificity

Sensitivity: Proportion of true positives; those who test positive and actually have the disease.
Specificity: Proportion of true negatives; those who test negative and do not have the disease.

Base Rate of Disease

Reflects the overall likelihood of having the disease without considering test results.

Positive Predictive Rate (PPR)

Probability of being diseased given a positive test result.

Bayes' Rule

Formula assists in reversing probability from a conditional perspective, allowing for reevaluation of likelihoods based on new evidence.

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Description

Test your understanding of the General Addition Rule in probability with these flashcards. This quiz covers key concepts such as disjoint events and conditional probabilities relevant to statistics. Perfect for students looking to reinforce their knowledge of this core topic.