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MODULE 4: PROBABILITY AND APPLICATION

Module 4.1: Probability

Module 4.1 Objectives

Students will be able to:o know the difference between Random

phenomenon and haphazard phenomenon. o understand the concept of probability as a

measure of how “likely” a specific event is happened.

o understand why probability is important for inferential statistics.

Review: Two Types of Statistics

Statistics

Descriptive Inferential

• Collecting• Organizing• Summarizing• Presenting

• Collecting• Organizing• Summarizing• Presenting

• Hypothesis Testing

• Determining Relationships

• Making predictions

• Hypothesis Testing

• Determining Relationships

• Making predictions

Probability is the basis for inferential statistics

Random Phenomenon

Random phenomenon has individual outcomes that are not completely predictable, but probabilities associated with the possible outcomes are well-defined.

Example: Flipping a coin

Haphazard phenomenon has individual outcomeswhere the probabilities associated with the possible outcomes are unknown.

Example: Asking someone to choose heads or tails

Definitions of Basic ConceptsDefinition Example

An experiment is a situation involvingchance or probability that leads to a result

The flipping of a coin

The result of an experiment is the outcome

Heads or tails.

An event is one or more outcomes of an experiment

A coin landing on heads in each of 3 trials is an event that contains one outcome (i.e., HHH). Two heads “HHT, HTH, THH” is an event which contains three outcomes.

All possible outcomes in anexperiment is the sample space

In an experiment flipping three coins (H = Head; T = Tail): {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

Example One: Coin Toss

Flipping a coin is a random phenomena because the probability of a specific outcome is well-defined.

We can conduct a coin flipping experiment to demonstrate this. This experiment would include flipping a coin over an over. Each coin flip is considered a trial of the experiment.

The outcome of each trial is well defined. It can be either heads or tails.

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Example One: Coin Toss (cont.)

An event is defined as a combination of outcomes from a random phenomenon that meet a specific criterion.

For example: The event that a coin lands on heads in each of 3 trials (i.e., HHH)

Trial 1 Trial 2 Trial 3

Coin flipping experiment with 3 trials

Why Probability? Probability is the basis for inferential statistics.

Everything is NOT predictable. o Predictions to test hypotheses are never exactly correct. Results

are only more or less likely.

Distinguish random vs. haphazard phenomena.o All science is based on observations and sample statistics that are

more or less affected by random variation.

Probability is important in public health because it helps us make predictions based on previous research.

o If an individual tests positive for HIV using the latest HIV test, what is the likelihood (or probability) that they actually are infected?

o What is the risk of developing type II diabetes among persons who are clinically obese?

What is Probability? Probability is a measure of how “likely” a specific event is

happened.

Probability is defined as the number of observations that satisfy some criterion divided by the total number of possible observations. Alternatively,

A basic definition of probability is the likelihood of an event being true divided by the total number of possibilities. For example, a fair coin with two sides.

Probability of head is P(head)=0.5

Overview of Basic Probabilityhttp://www.khanacademy.org/math/probability/v/basic-probability

Example One: Coin Toss (cont.)https://www.khanacademy.org/math/trigonometry/prob_comb/prob_combinatorics_precalc/v/coin-flipping-example

Probability Distribution

Two heads3/8

(0.375)

If we list all of the probabilities for a random phenomena we have created the probability distribution.

0.125+

One head3/8

(0.375)

No head1/8

(0.125)

Three heads

1/8(0.125)

0 ≤ Probability ≤ 1.0

0.1250.375+ 0.375+ =1.0

Probability distribution for three fair coins

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Basic Rules of Probability

The value of a probability must fall within the range from 0 to 1.0.o Probability of an impossible event (empty set ) is 0, P()=0.

o Probability of a certain event (sample space S) is 1, P(S)=1.

The sum of probabilities associated with all possible outcomes (events) for a random phenomenon must equal 1.0.

Example Two: Marble Bag

The marble bag: We have a bag with 9 red marbles, 2 blue marbles, and 3 green marbles in it.

What is the probability of randomly selecting a non-bluemarble from the bag?

Watch the video for details to calculate probability next slide.

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P(Non-blue marble) = 12/14 = 6/7

Example Two: Marble Bag (cont.)http://www.khanacademy.org/math/probability/v/probability-1-module-examples?exid=probability_1 Example Three: Playing Cards

Playing cards and Venn Diagramso We have an ordinary deck with 4 suits and each suit

has 13 types of cards A, 1, 2,….10, J, K, Q. − Probability of randomly selecting a Jack : P(Jack)=4/52=0.077− Probability of randomly selecting a Heart : P(Heart)= 13/52=0.25− Probability of randomly selecting a Jack and Heart: P(J and

H)=1/52=0.019− Probability of randomly selecting a Jack or Heart: P(J or

H)=16/52=4/13=0.308

oWatch the video next slide for the probability calculations.

Example Three: Playing Cards (cont.)http://www.khanacademy.org/math/probability/v/probability-with-playing-cards-and-venn-diagrams