Mind on Statistics (6th. Ed) Chapter 7 - Probability

Introduction

• Describe the relative frequency interpretation of probability, the personal probability interpretation, and situations where each interpretation is appropriate
• Recognise mutually exclusive and independent events
• Calculate probabilities using the addition and multiplication rules, including conditional probabilities
• Calculate probabilities using various tools, including tree diagrams, hypothetical hundred thousand table, and Bayes’ Rule
• Define vocabulary terms for medical screening tests, including sensitivity, specificity, positive predictive value, negative predictive value, disease prevalence, and disease incidence
• Demonstrate how to estimate probabilities using simulation
• Identify situations that produce flawed intuitive judgements about probability

Random Circumstances

Random Circumstance: is one in which the outcome is unpredictable (not determined until we observe it)

Interpretations of Probability

Relative

Relative Frequency Interpretation: applies to situations in which we can imagine repeatedly observing the results of a random circumstance.

Relative Frequency: For situations that we can imagine repeating many times, we define the probability of a specific outcome as the proportion of times it would occur over the long run. This also is called the relative frequency of that particular outcome.

The relative frequency probability is determined by either making an assumption about the physical world (e.g. coin toss) or making a direct observation of how often something happens.

Personal

Personal Probability: of an event is the degree to which an individual believes that the event will happen. Sometimes, the term subjective probability is used because the degree of belief may be different for each individual.

If the situation is not repeatable, we use personal otherwise relative.

Probability Definitions

Simple Event: a unique possible outcome of a random circumstance.

Sample Space: is the collection of all simple events for a random circumstance.

Compound Event: is an event that includes two or more simple events.

Event: is any collection of one or more simple events in the sample space and can be simple or compound. Often written as capital letters.

Complementary Event: if the two event do not contain any of the same simple events and together they cover the entire sample space. $A^C$ is the complement of $A$.

Mutually Exclusive/ Disjoint: if the two events do not contain any of the same simple events (outcomes). (does not cover the entire sample space)

Independent: if knowing one event will occur does not change the probability that the other occurs for two events.

• The physical situation makes it clear that knowing $A$ does not change probability $B$ will occur.
• If conditional probability is the same as unconditional probability
• $P(A \textrm{ and } B) = P(A)P(B)$

Conditional Probability: is the long-run relative frequency with which $B$ occurs when $A$ has occured or will occur. $P(B

A)$

Probability Rules

Complement Rule: \(P(A^C) = 1 - P(A)\)

Addition Rule (can extend to more than two events): \(P(A \textrm{ or } B) = P(A) + P(B) - P(A \textrm{ and } B)\\ P(A \textrm{ or } B, \textrm{ if mutually exclusive}) = P(A) + P(B)\\\)

Multiplication Rule (can extend to more than two events): \(P(A \textrm{ and } B) = P(A)P(B|A) = P(B)P(A|B)\\ P(A \textrm{ and } B, \textrm{ if independent}) = P(A)P(B)\)

\[P(\textrm{at least once}) = 1 - P(\textrm{none})\]

Language for Medical Screening Tests

Sensitivity: the probability of a postive test result when the patient has a disease

Specificity: the probability of a negative test result when the patient does not have the disease.

Positive Predictive Value (PPV): probability patient with a positive test result actually has the disease

Negative Predictive Value (NPV): probability patient with a negative test result does not have the disease

Prevalence: proportion of population that has the disease

Incidence: number/proportion of newly diagnosed cases of a disease in a specified time period.

Using Simulation to Estimate Probabilities

Simulate using a computer where the estimated probability is $\frac{x}{n}$ where $x$ is the relative frequency of event of interest for $n$ times. The conservative margin of error is $1/\sqrt{n}$.

Flawed Intuitive Judgements about Probability

• Confusion of the Inverse
• Specific People vs. Random Individuals
• Coincidences (a surprising concurrence of events, perceived as meaningfully related, with no apparent causal connection)
• Gambler’s Fallacy
• Law of Small Numbers