Chapter 5: Probability Concepts

1. Chapter 5: Probability Concepts

2. In Chapter 5: 5.1 What is Probability? 5.2 Types of Random Variables 5.3 Discrete Random Variables 5.4 Continuous Random Variables 5.5 More Rules and Properties of Probability

3. 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.

4. 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)

5. Probability as a repetitive process Experiments 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.

6. Subjective Probability Probability can be used to quantify a level of belief

7. §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…)

8. §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:

9. Discrete Random Variables The pmf can be shownin tabular or graphical form

10. 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.

11. Properties of ProbabilitiesIn 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)

12. Properties 1 & 2 Illustrated Property 1. 0 ≤ Pr(A) ≤ 1 Note that all individual probabilities are between 0 and 1. Property 2. Pr(S) = 1 Note that the sum of all probabilities = .0039 + .0469 + .2109 + .4219 + .3164 = 1 “Four patients” pmf

13. Ā A Property 3 Illustrated Property 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

14. A B Property 4 Illustrated Property 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

15. Area Under the Curve (AUC) “Four patients” pmf • 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

16. .2109 .0469 Example: Pr(X ≤ 2) = shaded “tail” = 0.0039 + 0.0469 + 0.2109 = 0.2617 .0039 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”)

17. §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

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

19. Examples of pdfs • pdfs obey all the rules of probabilities • pdfs come in many forms (shapes). Here are some examples: Uniform pdf Chi-square pdf Exercise 5.13 pdf Normal pdf The most common pdf is the Normal. (We study the Normal pdf in detail in the next chapter.)

20. 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.