1

Chapter 9

Introducing Probability

From Exploration to Inferencep. 150 in text

3

The Idea of Probability• Probability helps

us deal with “chance”

• Definition: the probability of an event is its expected proportion in an infinite infinite series of repetitions

Example: A random sample of n = 100 children has 8 individuals with asthma. What is the probability a child has asthma?

ANS: We do not know. Although 8% is a reasonable “guesstimate,” the true probability is not known because our sample was not infinitely large

4

How Probability BehavesCoin Toss Example

The proportion of heads approaches

0.5 with many, many tosses.

Chance behavior is unpredictable in the short run, but is predictable in the long run.

5

Probability models consist of these two parts:1)1) Sample SpaceSample Space (S) = the set of all possible

outcomes of a random process2)2) ProbabilitiesProbabilities (Pr) for each possible outcome in

the sample space

Probability Models

Example of a probability model

“Toss a fair coin once”

S = {Heads, Tails} all possible outcomes

Pr(heads) = 0.5 and Pr(tails) = 0.5 probabilities for each outcome

6

4 Basic Rules of Probability (Summary)

1.1. 0 0 ≤ Pr(A) ≤ 1≤ Pr(A) ≤ 1

2.2. Pr(S) = 1Pr(S) = 1

3.3. Addition Rule for Disjoint EventsAddition Rule for Disjoint Events

4.4. Law of ComplementsLaw of Complements

Also on bottom of page 1 of Formula Sheet

7

Rule 1 (Range of Possible Probabilities)

Let A ≡ event APr(A) ≡ probability of event A

Rule 1 says Rule 1 says ““0 0 ≤ Pr(A) ≤ 1≤ Pr(A) ≤ 1”” Probabilities are Probabilities are alwaysalways between 0 & 1 between 0 & 1

Pr(A) = 0 means A never occurs

Pr(A) = 1 means A always occurs

Pr(A) = .25 means A occurs 25% of the time

Pr(A) = 1.25 Impossible! Must be something wrong

Pr(A) = some negative number Impossible! Must be something wrong

8

Rule 2 (Sample Space Rule)Let S ≡ the Sample Space

Pr(S) = 1Pr(S) = 1

All probabilities in the sample space All probabilities in the sample space must sum to 1 exactly.must sum to 1 exactly.

Example: “toss a fair coin”

S = {heads or tails}

Pr(heads) + Pr(tails) = 0.5 + 0.5 = 1.0

9

Events A and B are disjoint if they can Events A and B are disjoint if they can never occur together. never occur together.

When events are disjoint:When events are disjoint:Pr(A or B) = Pr(A) + Pr(B)Pr(A or B) = Pr(A) + Pr(B)

Age of mother at first birthLet A ≡ first birth at age < 20: Pr(A) = 25%Let B ≡ first birth at age 20 to 24: Pr(B) = 33%Let C ≡ age at first birth ≥25 Pr(C) = 42%

Probability age at first birth ≥ 20 = Pr(B or C) = Pr(B) + Pr(C) = 33% + 42% = 75%

Rule 3 (Addition Rule, Disjoint Events)

10

Rule 4 (Rule of Complements)Let Ā ≡ A does NOT occur

This is called the complementcomplement of event A

Pr(Ā) = 1 – Pr(A)Pr(Ā) = 1 – Pr(A)

Example:

If A ≡ “survived” then Ā ≡ “did not survive”

If Pr(A) = 0.9 then Pr(Ā) = 1 – Pr(A) = 1 – 0.9 = 0.1

11

Probability Mass Functions pmfs

Example of a pmf: A couple wants three children. Let X ≡ the number of girls they will haveHere is the pmf that suits this situation:

Probability mass functions are made up of a list of separated outcomes.

For discrete random variables.

12

Probability Density Functions pdfs (“Density Curves”)

• To assign probabilities for continuous random variables we density curvedensity curve

• Properties of a density curve– Always on or above horizontal axis

– Has total area under curvearea under curve (AUCAUC) of exactly 1

– AUCAUC in any range = probability of a value in that range

Probability density functions form a continuum of possible outcomes.For continuous random variables.

13

Example of a pdf

Note• The curve is always on or above horizontal axis and has AUC =

height × base = 1 × 1 = 1• Probability = AUC in the range. Examples follow.• Pr(X < .5) = height × base = 1 × .5 = .5• Pr(X > 0.8) = height × base = 1 × .2 = .2 • Pr (X < .5 or X > 0.8) = .5 + .2 = .7

This random spinner has this pdf density “curve”

14

pdf Density Curves• Density curves come in many shapes

– Prior slide showed a “uniform” shape– Below are “Normal” and “skewed right” shapes

• Measures of center apply to density curves– µ (expected value or “mean”) is the center balancing point– Median splits the AUC in half

04/20/23 15

From Histogram to Density Curve

• Histograms show distribution in chunks

• The smooth curve drawn over the histogram represents a Normal density curve for the distribution

04/20/23 16

Area Under the Curve (AUC)

30% of students had scores ≤ 6

30% 70%

Area in Bars = proportion in that range30% of students

had scores ≤ 6

Shaded area = 30% of total area of the histogram

04/20/23 17

Area Under the Curve (AUC)

30%

Area Under Curve = proportion in that range!

30% of students had scores ≤ 6

30% of area under the curve (AUC) is shaded

70%

Summary of Selected Points

• To date we have studied descriptive statisticsdescriptive statistics. From here forward we study inferential statisticsinferential statistics {2}

• ProbabilityProbability is the study chance; chance is unpredictable in the short run but is predictable in the long run {3 - 4}; take the rules of probability to heart{5 - 10}

• Discrete random variablesDiscrete random variables are described with probability mass function

• Continuous random variablesContinuous random variables are described with density curves density curves with the area under the curve curve (AUC)(AUC) corresponding to probabilities