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Lesson 9-4 Homework And Practice Theoretical Probability Definition

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Theoretical Probability versus Experimental Probability

Theoretical Probability versus Experimental Probability

You've heard the terms, theoretical probability and experimental probability. but what do they mean? Are they in anyway related? This is what we are going to discover in this lesson.

If you've completed the lessons on independent and dependent probability. then you've already found the theoretical probability for numerous problems.

Theoretical Probability

Theoretical probability is the probability that is calculated using math formulas. This is the probability based on math theory.

Experimental Probability

Experimental probability is calculated when the actual situation or problem is performed as an experiment. In this case, you would perform the experiment, and use the actual results to determine the probability.

In order to accurately perform an experiment, you must:

  • Identify what constitutes a "trial ".
  • Perform a minimum of 25 trials
  • Set up an organizer (table or chart) to record your data.

Let's take a look at an example where we first calculate the theoretical probability, and then perform the experiment to determine the experimental probability. It will be interesting to compare the theoretical probability and the experimental probability. Do you think the two calculations will be close?

This problem is from Example 1 in the independent events lesson. We calculated the theoretical probability to be 1/12 or 8.3%. Take a look:

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Theoretical and Experimental Probability - Definition & Examples

Theoretical and Experimental Probability

Probability theory is an essential subject of mathematics. We study about probabilities in this subject.

The probability may be defined as chances of an event to be happen. It is the numerical measure of the extent of likeliness of occurrence of some event. In other words, to find p robability of an event is to quantify the chances of its happening or not happening.

The probability lies between 0 and 1. The event whose probability is 0 is an impossible event ; while an event with probability 1 is said to be a sure event. Probability can be expressed in the form of fractions and decimals.
There are mainly two types of probabilities:
i)Theoretical Probability
ii)Experimental Probability As the names themselves suggest that the theoretical probability is based upon theory or formula. On the other hand, the experimental probability is calculated by the actual experiments. Let us go ahead and learn these two types of probabilities and methods of evaluating them in detail.

Definition

Theoretical Probability
The theoretical probability is the measure of theoretically happening of an event. This is something which is expected to be happened. Theoretical probability is calculated in a theoretical way. It is considered as the standard. It is not necessary that this probability actually occurs every time. Theoretical probability may differ from the actual occurrence of the event. This probability is estimated using formula. This can be done by dividing the possible number of favorable outcomes by the total outcomes.

Theoretical probability denotes the possible number of ways by which an event can occur. This probability is a method of determining what could happen in a situation, on the basis of given information. It is mere calculation.

By theoretical probability, we may have the idea of something which is likely to occur in a particular situation, but the actual result cannot be predicted by this.

It is calculated by the following formula:

There is another way of finding probability of some event. It is to perform an experiment. The e xperimental probability is the probability of some event which occurs during an actual experiment. Experimental probability can be determined by conducting the experiment. It is not just estimated mathematically.

We have discussed that the t heoretical probability is something that is expected to be occur; while experimental probability is the measure what actually occurs on an experiment. Experimental probability is also calculated by the similar formula described above. But the number of favorable outcomes are found by the real conduction of experiment. I n experimental probability, t he whole event should be performed.

The main difference between theoretical probability and experimental probability is that t heoretical probability is a measure of what should happen; on the other hand, experimental probability is the measure what had happened in actual practice.

For Example:
The theoretical probability of obtained any number on a roll of die is $\frac<1><6>$. This should ideally happen. But in an actual practice, maybe the die gets that number 2-3 times in successive trials. This may be different than the theoretical probability.

How to find Theoretical Probability

In order to find theoretical probability, one should follow the steps mentioned below:

Step 1: Firstly observe the event closely.

For Example: Let the event be flip of a coin. It should be kept in mind that a coin has two faces - one is called head and another is known as tails.

Step 2: Focus on what to be determined. Let us suppose that in this example, the chances of appearing a tail is asked.

Step 3: Calculated the total number of possible outcomes. Here, the total outcomes would be 2 - head and tail.

Step 4: Now determine theoretically the number of favorable outcomes. In this case, on a coin, favourable outcome is 1 i.e. tail itself.


Step 5: Substitute these values in the formula:

In this example:


The theoretical probability of getting heads and tails are 50% each. In this case, the experimental probability of heads is slightly higher than theoretical one and that of tails is slightly less that theoretical probability.

Example2: Two dice are rolled 50 times and their outcomes are noted as follows :

Sum of the outcomes of both dice are

3, 4, 6, 5, 5, 8. 9, 10, 7, 5, 12, 9, 6, 5, 7, 8, 7, 4, 8, 8, 11, 6, 8. 6, 8, 4, 4, 5, 7, 8, 9, 7, 8, 7, 8, 11, 6, 7, 5, 4, 3, 6, 7, 7, 8, 9, 7, 8, 6, 7

1) Calculate the experimental probability of getting a sum of 8.

2) Find the theoretical probability of getting a sum of 8.

3) Compare both the probabilities.

Statistics 7 - Winter 2016

Statistics 7: Basic Statistics Winter 2016
University of California, Irvine ANNOUNCEMENTS

GRADE DISPUTE REQUEST FORM(pdf ) . If you would like to submit a request for a grade dispute on any of the homeworks or midterm exams, download and fill out this form (read the directions carefully!), then turn it in to Prof. Hancock along with the homeworks and/or midterms that you are contesting any time before Monday, March 14 at 5:00pm . You must turn in the original homework/exam with the grade dispute request form, so if you need the homework/exam to study for the final exam, make a copy before turning it in. Grade dispute forms and the homework/exam under dispute will not be returned to you.

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Course Syllabus

Download the course syllabus here .

Textbook Information

For the book companion website (including supplemental videos and applets by chapter), click here.
For textbook options including online versions, rentals, and using the previous editions, click here .

There are three copies of our textbook on reserve in the Ayala Science Library, available for check-out for a period of 2 hours.

Distribution Tables from our Textbook
  • Table A.1. Cumulative probabilities (area under the curve to the left of z ) of a standard normal distribution
  • Tables A.2 and A.3. Percentiles (t* ) and (1 - cumulative probabilities) (area under the curve to the right of t ) for t-distributions
Lecture and Discussion Locations and Times R Commander Resources
Installing R and R Commander and Getting Started Access R and R Studio through UCI Virtual Computer Lab (VCL)
You may access R and R Studio without downloading the software through UCI's Virtual Computer Lab: http://www.oit.uci.edu/labs/vcl/. To use this resource,
  1. Click on "Click Here to Start Using VCL".
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  3. Click "New Reservation" in the left menu.
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More information for Mac users can be found here .

Information on UCI Computer Labs (with R Commander installed)
R Commander Handouts and Instruction Sheets After installing R and following the instructions above to install R Commander, to open R Commander, first open R, then in the R console (after the ">"), type
R is case sensitive, so make sure you have the capital "R" followed by lowercase "cmdr".
  • R Commander Instruction Sheet (Week 1 Discussion)
  • Chi-squared test in R Commander (Section 4.4)
  • Calculating binomial probabilities in R Commander (Section 8.4)
Data sets used in lecture/discussion or on homework:

All data sets are in "comma separated value" (.csv) format. When importing into R Commander from a URL, make sure you choose "Commas" for the Field Separator. Copy and paste the URL below into R Commander (Mac users: Use keyboard shortcut Control-V to paste rather than the usual Command-V.)

staceyah/7/data/Ex2.2.csv
Penn State student data (Example 2.5): URL http://www.ics.uci.edu/

staceyah/7/data/pennstate1.csv
CEO Data (Example 2.6): URL http://www.ics.uci.edu/

staceyah/7/data/ceodata08.csv
Manatee Data. URL http://www.ics.uci.edu/

staceyah/7/data/Manatees.csv
SAT Data (Homework 2): URL http://www.ics.uci.edu/

staceyah/7/data/sats98.csv
Speed Limit Data (Homework 2): URL http://www.ics.uci.edu/

staceyah/7/data/speedlimit.csv
Cholesterol Data (Homework 2): URL http://www.ics.uci.edu/

staceyah/7/data/cholesterol.csv
EEE Stats 7 Survey Data (Homework 7): URL http://www.ics.uci.edu/

Class Calendar

This calendar will be updated after each class. Check back periodically for required reading, homework assignments, lecture slides, and extra resources.

Reading Assignment and Material Covered
(Tentative schedule; may be updated after each class)

Sample Exam Problems

Both midterm exams and the final exam will be a mix of multiple choice questions and of free-response questions. Below are examples of both types of problems. Note that there are more sample problems than the number of questions that will actually be on the exam to allow for more practice. Also, much more space would be provided on the actual exam for you to answer the questions. Do not waste this resource! These practice problems will not help you if you look at the solutions before attempting the problems.

Sample exam problems will be posted one week prior to each exam.

Midterm Exam 1 Midterm Exam 2 Final Exam

Our final exam is cumulative. Practice problems from Chapters 1-8 were posted with the midterm exams. Practice problems for Chapters 9-13 are below.

Homework and Midterm Exam Solutions

Homework solutions and midterm exam solutions, including instructions on how the exam was graded and common student mistakes, will be posted in our "Stats 7 Solutions" EEE Dropbox. Sign into EEE to access this resource.

Ed602 Lesson 7

Definition Page. Contains definitions arranged alphabetically.

Notes:
So far we have been plotting and analyzing distributions that come from actual data that we have collected. In the notes, the homework and the quizzes, we have real scores from real samples. These distributions are called empirical distributions. There is another set of distributions called theoretical distributions. These are hypothetical distributions that only appear under certain unusual circumstances that may or may not actually exist. This chapter, on the standard normal curve, and standard scores is based on the information you can get from these theoretical distributions.

The normal distribution

A symmetrical, continuous, and asymptomatic bell shaped distribution of scores

Scores bunch up around the mean

The mean, median, and mode are all the same number

The normal curve

A graph of the normal distribution

The curve never actually touches the x-axis

When graphed, it generally shows 6-8 standard deviations (3-4 one either side of the mean)

If we divide the distribution up into standard deviation units, a known proportion of scores lies within each portion of the curve.

Tables exist so that we can find the proportion of scores above and below any part of the curve, expressed in standard deviation units. Scores expressed in standard deviation units, as we will see shortly, are referred to as Z-scores.

Area under the normal curve

Looking at the figure above, we can see that 34.13% of the scores lie between the mean and 1 standard deviation above the mean. An equal proportion of scores (34.13%) lie between the mean and 1 standard deviation below the mean. We can also see that for a normally distributed variable, approximately two-thirds of the scores lie within one standard deviation of the mean (34.13% + 34.13% = 68.26%).

13.59% of the scores lie between one and two standard deviations above the mean, and between one and two standard deviations below the mean. We can also see that for a normally distributed variable, approximately 95% of the scores lie within two standard deviations of the mean (13.59% + 34.13% + 34.13% + 13.59% = 95.44%).

Finally, we can see that almost all of the scores are within three standard deviations of the mean. (2.14% + 13.59% + 34.13% + 34.13% + 13.59% + 2.14% = 99.72%) We can also find the percentage of scores within three standard deviation units of the mean by subtracting .13% + .13% from 100% (100.00% - (.13% + .13%) = 99.74%). (The difference in these totals 99.72, and 99.74 is due to rounding)
Standard normal distribution (aka, unit normal distribution)

Has a mean of 0

Has a standard deviation of 1

Scores from this distribution are called z scores

Standard scores (z scores)

Changing raw scores to z scores allows for comparisons across measurement tools that are not already equivalent

It is only possible to change raw scores to z scores if the raw scores come from a relatively normally distributed set of scores

  • Positive z scores indicate a score above the mean, negative z scores indicate a score below the mean. A z score of zero indicates a score at the mean.
  • Standard score formula

    A z of –1.03 is paired with a proportion of .3485 between the score and the mean. Because we are below the mean, we subtract this number from 0.5000 (which is the mean) to get a proportion of 0.1515. a score of 40 is at the 15.15th percentile.

    Normal distribution – a theoretical mathematical distribution that specifies the relative frequency of a set of scores in a population.

    Standard normal distribution – A normal distribution with a mean of 0 and a standard deviation of 1.

    Outcome – each possible occurrence in a probability distribution

    Event – the occurrence of a specific set of outcomes in a probability distribution

    Probability of occurrence of an event – the number of outcomes comprising an event, divided by the total number of possible outcomes

    Discrete outcomes – Outcomes in a distribution that have a countable set of outcomes

    Mutually exclusive outcomes – outcomes that cannot occur at the same time

    Theoretical probability distribution – a probability distribution found from the use of a theoretical probability model

    Empirical probability distribution – a probability distribution found by counting actual occurrences of an event

    Standard score – A score obtained by using the transformation z = X - / S

    Lesson 11 3 theoretical probability

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