New version page

MC MA 116 - Chapter 4 - Fundamentals of Probability

Upgrade to remove ads
Upgrade to remove ads
Unformatted text preview:

To evaluate the mean and standard deviation using the calculatorFind binomial probabilities with a shortcut feature of the calculatorChapter 4 - Fundamentals of Probability Experiment: Any process that allows researchers to obtain observations Sample Space: All possible outcomes of an experiment Simple Event: Consists of a single outcome of an experiment Event: Consists of one or more outcomes of an experimentNotation for ProbabilitiesProbability of Event A is denoted P(A)Round-Off-Rule for Probability: Use 3 significant digits as decimals (or use fraction form).Finding Probabilities with the Classical Approach (Requires Equally Likely Outcomes) method. . . . .( ). . . .number of ways A can occur sP Anumber of different simple events n= =Probability Values▪ For any event A, 0 ( ) 1P A� �▪ The probability of an impossible event is zero▪ The probability of a certain event is oneComplementary EventsThe complement of event A, denoted byA (other books may use A’ or CA), consists of all simple outcomes in the sample space not making up event A.Rule of Complementary EventsSince P(A) + P(A) = 1then P(A) = 1 – P(A) and P(A) = 1 – P(A)1Chapter 5 – Probability Distributions, Probability HistogramsA random variable is a variable (typically represented by x) that has a numeric value, determined by chance, for each possible outcome of an experimentExamples:The number of students passing a certain class The average height of the students in a class The number of girls in a family of 5 children The sum on the faces of two rolled dice The number of defective parts in a sample of 20 The average daily temperatureA word about randomnessThe word randomness suggests unpredictability.Randomness and uncertainty are vague concepts that deal with variation.A simple example of randomness involves a coin toss. The outcome of the toss is uncertain. Since the coin tossing experiment is unpredictable, the outcome is said to exhibit randomness.Even though individual flips of a coin are unpredictable, if we flip the coin a large number of times, a pattern will emerge. Roughly half of the flips will be heads and half will be tails. This long-run regularity of a random event is described with probability. Our discussions of randomness will be limited to phenomenon that in the short run are not exactly predictable but do exhibit long run regularity.A discrete random variable has either a finite or a countable number of values. This chapter deals with discrete random variables. A continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale in such a way that there are no gaps or interruptions. A probability distribution is a graph, table, or formula that gives the probability for each possible value of the random variable. (Notice: similar to relative frequency tables, histograms)A probability histogram is a way to graph a probability distribution.The vertical scale shows probabilities instead of relative frequencies. Note that the area of these rectangles is the same as the probabilities.2- Requirements for a Probability Distributiono 0  P(X = x)  1o The sum of the probabilities of a discrete random variable is 1.( ) 1P X x - Using the calculator to find the mean (expected value) and standard deviation for a probability distributionEnter x into L1Enter the probabilities into L2Press STATArrow right to CALCSelect 1: 1-Var Stats L1,L2Press ENTER- Identifying Unusual Results with the Range Rule of Thumb Minimum usual value = 2m s-Maximum usual value = 2m s+- Identifying Unusual Results with the Probability RuleUnusually high: x successes among n trials is unusually high if P(x or more) is very small (suchas less than 0.05)Unusually low: x successes among n trials is unusually low if P( or fewer) is very small (such asless than 0.05)31) EXPERIMENT: Rolling a 1-6 die and recording the number obtained a) What is the sample space? These are the possible values of the random variable.b) What is the probability of rolling the number 6?c) What is the probability of not rolling the number 6?d) What is the probability of rolling the number 7?e) What is the probability of rolling a number less than 7?f) What is the probability of obtaining a number less than 2 or a number greater than or equal to 5?g) Complete the following table which is the probability distribution for this experiment. Sketch the probability histogram and describe the shape of the distribution.Outcome probability123456h) Refer to the Probability Rule given below and answer the question: are there any unusual outcomes on this experiment?- Identifying Unusual Results with the Probability RuleUnusually high: x successes among n trials is unusually high if P(x or more) is very small (such as less than 0.05)Unusually low: x successes among n trials is unusually low if P( or fewer) is very small (such as less than 0.05)4Experimental Probability Method . . . .( ). . . . .number o times A occurredP Anumber of times trial was repeated=i) Simulate the experiment of rolling a die 10 times by using the calculator. (Calculator instructions to generate 10 random integers from 1 to 6:MATH PRB 5:randInt(1,6,10) ENTERj) Use your results to find the experimental probability of rolling the number 6k) How does your answer to (j) compare to the answer to (b)?Part (b) gives the theoretical probability, what we expect to happen, and part (j) give the experimental probability, which is what actually happened in the experiment. Do you have any idea in what to do in order to get the experimental probability closer to the theoretical probability? l) Let’s collect data from all students in the classroom (we’ll do this in class)OutcomexTally Frequency Relative FrequencyProbabilitym) How does your answer to (l) compare to the answer to (b)?Law of Large Numbers: As a procedure is repeated again and again, the relative frequency probability of an event is expected to approach the actual theoretical probability.52) Use the calculator to find the mean and standard deviation of a probability distributionUse the class results from the previous page to complete the probability distribution for the simulation of the experiment of rolling a die and recording the outcomeOutcome probability123456- To evaluate the mean and standard deviation using the calculatorEnter x into L1Enter the probabilities into L2Press STATArrow right to CALCSelect 1: 1-Var Stats


View Full Document
Download Chapter 4 - Fundamentals of Probability
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Chapter 4 - Fundamentals of Probability and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Chapter 4 - Fundamentals of Probability 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?