May 3, 2023

Chapter 5: Chapter 5: Probability ConceptsProbability Concepts

In Chapter 5:

5.1 What is Probability?5.2 Types of Random Variables5.3 Discrete Random Variables 5.4 Continuous Random Variables5.5 More Rules and Properties of Probability

Definitions• Random variable ≡ a numerical quantity that

takes on different values depending on chance• Population ≡ the set of all possible values for a

random variable• Event ≡ an outcome or set of outcomes• Probability ≡ the proportion of times an event is

expected to occur in the population

Ideas about probability are founded on relative frequencies (proportions) in populations.

Probability Illustrated• In a given year, there were 42,636 traffic

fatalities in a population of N = 293,655,000 • If I randomly select a person from this

population, what is the probability they will experience a traffic fatality by the end of that year?

ANS: The relative frequency of this event in the population = 42,636/ 293,655,000 = 0.0001452. Thus, Pr(traf. fatality) = 0.0001452 (about 1 in 6887 1/.0001452)

Probability as a repetitive processExperiments sample a population in which 20% of observations are positives. This figure shows two such experiments. The sample proportion approaches the true probability of selection as n increases.

Subjective ProbabilityProbability can be used to quantify a level of

beliefProbability Verbal expression

0.00 Never0.05 Seldom

0.20 Infrequent

0.50 As often as not0.80 Very frequent

0.95 Highly likely1.00 Always

§5.2: Random Variables• Random variable ≡ a numerical quantity that

takes on different values depending on chance• Two types of random variables• Discrete random variables: a countable set of

possible outcome (e.g., the number of cases in an SRS from the population)

• Continuous random variable: an unbroken continuum of possible outcome (e.g., the average weight of an SRS of newborns selected from the population (Xeno’s paradox…)

§5.3: Discrete Random Variables• Probability mass function (pmf) ≡ a

mathematical relation that assigns probabilities to all possible outcomes for a discrete random variables

• Illustrative example: “Four Patients”. Suppose I treat four patients with an intervention that is successful 75% of the time. Let X ≡ the variable number of success in this experiment. This is the pmf for this random variable:

x 0 1 2 3 4Pr(X=x) 0.0039 0.0469 0.2109 0.4219 0.3164

Discrete Random Variables The pmf can be shown in tabular or graphical form

x 0 1 2 3 4Pr(X=x) 0.0039 0.0469 0.2109 0.4219 0.3164

Properties of Probabilities• Property 1. Probabilities are always between 0

and 1• Property 2. A sample space is all possible

outcomes. The probabilities in the sample space sum to 1 (exactly).

• Property 3. The complement of an event is “the event not happening”. The probability of a complement is 1 minus the probability of the event.

• Property 4. Probabilities of disjoint events can be added.

Properties of Probabilities In symbols

• Property 1. 0 ≤ Pr(A) ≤ 1• Property 2. Pr(S) = 1, where S represent

the sample space (all possible outcomes)• Property 3. Pr(Ā) = 1 – Pr(A), Ā represent

the complement of A (not A)• Property 4. If A and B are disjoint, then

Pr(A or B) = Pr(A) + Pr(B)

Properties 1 & 2 IllustratedProperty 1. 0 ≤ Pr(A) ≤ 1Note that all individual probabilities are between 0 and 1.

Property 2. Pr(S) = 1Note that the sum of all probabilities = .0039 + .0469 + .2109 + .4219 + .3164 = 1

“Four patients” pmf

Property 3 IllustratedProperty 3. Pr(Ā) = 1 – Pr(A),

As an example, let A represent 4 successes.

Pr(A) = .3164

Let Ā represent the complement of A (“not A”), which is “3 or fewer”.

Pr(Ā) = 1 – Pr(A) = 1 – 0.3164 = 0.6836

“Four patients” pmf

Ā

A

Property 4 IllustratedProperty 4. Pr(A or B) = Pr(A) + Pr(B) for disjoint events

Let A represent 4 successes Let B represent 3 successes

Since A and B are disjoint, Pr(A or B) = Pr(A) + Pr(B) = 0.3164 + 0.4129 = 0.7293.

The probability of observing 3 or 4 successes is 0.7293 (about 73%).

“Four patients” pmf

B A

Area Under the Curve (AUC)• The area under

curves (AUC) on a pmf corresponds to probability

• In this figure, Pr(X = 2) = area of shaded region = height × base = .2109 × 1.0 = 0.2109

“Four patients” pmf

Cumulative Probability• “Cumulative

probability” refers to probability of that value or less

• Notation: Pr(X ≤ x) • Corresponds to AUC to

the left of the point (“left tail”)

.0469

.2109

.0039

Example: Pr(X ≤ 2) = shaded “tail” = 0.0039 + 0.0469 + 0.2109 = 0.2617

§5.4 Continuous Random Variables

Continuous random variables form a continuum of possible values. As an illustration, consider the spinner in this illustration. This spinner will generate a continuum of random numbers between 0 to 1

§5.4: Continuous Random Variables

A probability density functions (pdf) is a mathematical relation that assigns probabilities to all possible outcomes for a continuous random variable. The pdf for our random spinner is shown here.

The shaded area under the curve represents probability, in this instance: Pr(0 ≤ X ≤ 0.5) = 0.5

0.5

Examples of pdfs• pdfs obey all the rules of probabilities• pdfs come in many forms (shapes). Here are

some examples:

Uniform pdf Normal pdf Chi-square pdf Exercise 5.13 pdf

The most common pdf is the Normal. (We study the Normal pdf in detail in the next chapter.)

Area Under the Curve• As was the case with pmfs,

pdfs display probability with the area under the curve (AUC)

• This histogram shades bars corresponding to ages ≤ 9 (~40% of histogram)

• This shaded AUC on the Normal pdf curve also corresponds to ~40% of total.

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