Describe How to Use Technology.

Textbook: Statistics Informed Decisions Using Data; 4th edition, Sullivan III, M.

Unit 3 Question 1: Probability in Real Life

What method of assigning probability (relative frequency, classical, or subjective) discussed in Unit 3 do you feel is the most appropriate to use and why? In your response, please provide an example that is not found in your text. Please remember to cite all resources used and use your own words in your explanation.

Unit 3 How to Use Technology
Below, you will find directions on how to use the available technology tools to generate a random number if desired for practice.

NOTE: Special Technology is not necessary in this unit as you will be provided Random Number tables in each question.

Reading Supplement:

MAT220 Technology Step By Step CH 5.docx
UNIT 3 – Chapter 5 – Overview
PROBABILITY

In the previous two units we developed the fundamental tools of descriptive statistics. Unit 3 deals with the basic theory and concepts of probability. Probability, in combination with the descriptive techniques we have learned, allows us to proceed into inferential statistics in later chapters.

All of you have at some time in your life been exposed to situations where you wondered what the likelihood of something happening was. For example, if you were planning a trip to the beach, you might ask yourself whether or not it would rain. Weather forecasters make predictions all the time on the likelihood of a weather event occurring. In this unit, we will discuss how probability is assigned to an event and give you a better understanding of how probability can be a useful decision-making tool.

Many events exhibit a random character, yet by repeated observations of these events, we can see long term patterns that persist despite short term fluctuations. The result of a probability statement is the likelihood that an event will occur. In probability, we are concerned with drawing inferences about something with a level of certainty.

Probability can be expressed as a system of definitions and operations pertaining to a sample space. Every probability statement is related to a sample space of some sort. Take for example throwing a single die. The sample space for performing this act is {1, 2, 3, 4, 5, 6}. From this sample space, when the die is thrown, we will obtain one of six possible numbers.

Read the following definitions and develop an understanding of how they relate to the discussion of probability.

Statistical Experiment: A process that generates a set of data. Such a process leads to a myriad of results, each with some possibility of occurring.
Sample Space: A set of all possible outcomes of a statistical experiment.
Sample Point: Each possible outcome of a statistical experiment, called a sample point, is an element of the sample space. An example of this would be the Queen of Diamonds in a 52-card deck of cards.
Event: A subset of the sample space. It is an observable happening. There are two types of events:
Simple: Contains only one sample point.
Compound: Contains two or more sample points.

Example:

A Census is taken to determine the # of persons in selected households. The sample space S = {the # of persons in households}. If we define Event A = {4} and Event B = {x ≥ 5}, then Event A is a simple event, since it consists of a single outcome (namely, that there are 4 persons in a household), while Event B is a compound event, since it comprises multiple outcomes (namely, that there are 5, or 6, or 7, or 8, or 9, … persons in a household).

Null Set: A set with no sample points (also known as the empty set).
A statement of probability is made about the relative frequency of an event that is associated with a sample space.

Let A, B, and C stand for Events. The probability of an event A will be denoted by P(A).

The probability of an event A, P(A), is the ratio of the number of sample points that are examples of A to the total number of sample points in the sample space, assuming all points are equally likely.

Example:

Let A be the event “3” when a die is cast. How many sample points are examples of the event? The answer is 1. What is the total number of sample points? The answer is 6. The probability of event A (“3”) then is the number of examples of A over the total number of sample points:
P(A) =

Example:

What is the probability of getting a queen in a deck of cards? The number of examples of A is 4 and the total number of sample points is 52:

P(A) = =

There are three counting rules that are important in assigning probability. Being able to identify and count the number of experimental outcomes is a necessary step to assigning probabilities:

1. Multiple Step Experiments rule – Makes it possible to determine the number of experimental outcomes without listing them.

Example: Let’s say we are tossing 2 coins.

The experimental outcomes are defined in terms of the pattern of heads and tails appearing in the upward faces of the 2 coins.

Step 1 – Tossing the 1st coin
Step 2 – Tossing the 2nd coin

The sample space for this experiment is: {(H, H), (H, T), (T, H), (T, T)}

There are 4 experimental outcomes:
Since there are 2 possible outcomes in the first step, and for each of those there are 2 possible outcomes in the second step,

(2 steps) x (2 outcomes) yields (2) (2) = 4 distinct outcomes

What would be the # of experimental outcomes when tossing 6 coins?

(6 steps) x (2 outcomes) yields (2) (2) (2) (2) (2) (2) = 64 distinct outcomes

2. Combinations rule – Allows the counting of the # of experimental outcomes when n objects are to be selected from a set of N objects:

Defined as the number of Combinations of N objects taken n at a time:

! means Factorial; for example: 5! = (5) (4) (3) (2) (1) = 120.

Other notations for combinations include and .

Example:

Consider a Quality Control Inspector who randomly selects 2 of the 5 parts to test for defects. In a group of 5 parts, how many combinations of 2 can be selected? Here, N = 5 and n = 2.

There are 10 outcomes for the experiment.

3. Permutations rule – Allows one to compute the number of experimental outcomes when n objects are to be selected from a set of N objects where the order of selection is important.

Using the numbers from the previous example:

There are 20 outcomes for the experiment of randomly selecting 2 parts from a group of 5 when the order of selection must be taken into account.

How Probability is Assigned

There are three approaches to assigning probabilities to experimental outcomes. There are two basic requirements that must be satisfied, regardless of the approach used:

The probability assigned to each outcome must be between 0 and 1.
The sum of all the probabilities for all of the experimental outcomes must equal 1.

A. Classical – Appropriate when all the experimental outcomes are equally likely. If there are n experimental outcomes, a probability of is assigned to each experimental outcome. Both requirements are satisfied under this approach automatically. For example, when rolling a single die, there are 6 possible outcomes. The probability assigned to each possible outcome from rolling a die is , which is between 0 and 1. The sum of probabilities for all possible outcomes from rolling a die is equal to 1 (since +++++= 1).

B. Relative Frequency – Appropriate when data is available to estimate the proportion of time the experimental outcome will occur when the experiment is repeated a large number of times.

Example:

Let’s say we wanted to study the number of patients waiting for service in the X-ray unit of a hospital. Over a 20 consecutive day period, the number of patients that were found waiting at 9:00 a.m. is as follows:

From the above table, we can see that there were no people waiting on 2 of the 20 days and can assign a probability of = .10 to this experimental outcome. Probabilities for the other numbers of waiting patients are assigned analogously.

As with the Classical Method, both requirements are satisfied automatically. Probability of each outcome is between 0 and 1 and the sum of all probabilities is equal to 1.

C. Subjective Method – Most appropriate when it is unrealistic to assume that the experimental outcomes are equally likely and when little relevant data is available. Use any information available such as your experience or intuition.

For example, let’s assume that you and a friend have made an offer to purchase a home. Two outcomes are possible: the offer is accepted or the offer is rejected.

You believe that the offer will be accepted with a .8 probability and that it will be rejected with a .2 probability. Your friend believes that the offer will be accepted with a .6 probability and that it will be rejected with a .4 probability.

Both you and your friend have assigned probabilities that satisfy the two requirements. The fact that they differ emphasizes the personal nature of the subjective method.

Conditional probability is the probability of one event occurring given that another has taken place. The vertical bar notation is used in conditional probability to denote that we are considering the probability of an event given the condition that another event has occurred. Accordingly, the notation P(A|B) is read as “the probability of A given B”. Please read Section 5.4 in your text and watch the video in Blackboard for additional information on conditional probability.