PART IV : PROBABILITY - Binghamton University

[Pages:91]Chapter 13 & 14 - Probability

PART IV : PROBABILITY

Dr. Joseph Brennan

Math 148, BU

Dr. Joseph Brennan (Math 148, BU)

Chapter 13 & 14 - Probability

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Why Should We Learn Probability Theory?

Dr. Joseph Brennan (Math 148, BU)

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Let's Make a Deal!

Dr. Joseph Brennan (Math 148, BU)

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Uncertainty

Usually the results of a study, observational or experimental, are uncertain: if we repeat a study, we will not get exactly the same results. Example 1 (A coin). You toss a coin. Will it land heads or tails? Example 2 (A die). A regular die, a D6, is a cube with six faces:

What number will a die show when it is rolled once?

Dr. Joseph Brennan (Math 148, BU)

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Uncertainty

Example 3 (Box with marbles) A box contains 7 marbles: 2 red and 5 green. Each marble has an equal chance to be selected. One marble is to be drawn at random from the box. What color will it be?

Example 4 (Box with tickets) A box contains 10 tickets labeled 1 through 10. What will be the number of a randomly selected ticket?

Example 5 (Height of students) Seven students will be selected at random from the Math 148 class list and their heights will be measured. What will be the average height? Will it change if we choose different seven students?

Dr. Joseph Brennan (Math 148, BU)

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Probability

What is the chance or probability that we will make the same conclusions every time when we replicate a study? What is the chance or probability that the histogram will change? Knowledge of probability theory will help us answer these questions.

NOTE: In this part, the words chance and probability will have the same meaning.

chance = probability

Dr. Joseph Brennan (Math 148, BU)

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Inferential Statistics & Probability Theory

In Parts II and III we used random samples to collect evidence and make inferences about population.

Sample mean, x?, estimates unknown population mean ?.

Sample standard deviation, s, estimates unknown population standard deviation .

Regression equation is a mathematical model which approximates the true relationship between x and y .

In this part we will think in the opposite direction; we will reason from a known population to randomly selected samples.

Probability Theory

Inferential Statistics

P opulation

Sample

Sample

P opulation

Dr. Joseph Brennan (Math 148, BU)

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Outcomes and Sample Space

Probability theory deals with studies where the outcomes are not known for sure in advance. Usually, there are many possible outcomes for a study, we just do not know which particular outcome we will observe.

Sample Space: The set of all possible outcomes of a study. The

sample space of a study is denoted by S.

Every repetition of a study, or a trial, produces a single outcome. Usually an outcome is computed from the values of the response variables.

In Example 5 (Height of students) the outcome is the average height, y?, which is computed from the values of the response variable (y a student's height).

Conclusions from a study are based on its outcome.

Dr. Joseph Brennan (Math 148, BU)

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