Download To evaluate the mean and standard deviation using

Chapter 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 experiment
Notation for Probabilities
Probability 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
P( A) 
number.of .ways. A.can.occur
s

number.of .different.simple.events n
Probability Values
▪ For any event A, 0  P( A)  1
▪ The probability of an impossible event is zero
▪ The probability of a certain event is one
Complementary Events
C
The complement of event A, denoted by A (other books may use A’ or A ), consists of all simple
outcomes in the sample space not making up event A.
Rule of Complementary Events
Since P(A) + P( A ) = 1
then
P(A) = 1 – P( A )
and P( A ) = 1 – P(A)
1
Chapter 5 – Probability Distributions, Probability Histograms
A random variable is a variable (typically represented by x) that has a numeric value, determined by chance, for
each possible outcome of an experiment
Examples:
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 temperature
A word about randomness
The 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 Distribution
o
o
0  P(X = x)  1
The sum of the probabilities of a discrete random variable is 1.
 P( X  x)  1

Using the calculator to find the mean (expected value) and standard deviation for a probability
distribution
Enter x into L1
Enter the probabilities into L2
Press STAT
Arrow right to CALC
Select 1: 1-Var Stats L1,L2
Press ENTER

Identifying Unusual Results with the Range Rule of Thumb
Minimum usual value =
Maximum usual value =

  2
  2
Identifying Unusual Results with the Probability Rule
Unusually 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)
3
1) 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 probability
1
2
3
4
5
6
h) 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 Rule
Unusually 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)
4
Experimental Probability Method
P( A) 
number.o.times. A.occurred
number.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)
ENTER
j) Use your results to find the experimental probability of rolling the number 6
k) 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)
Outcome
x
Tally
Frequency
Relative
Frequency
Probability
m) 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.
5
2) Use the calculator to find the mean and standard deviation of a probability distribution
Use 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 outcome
Outcome probability
1
2
3
4
5
6

To evaluate the mean and standard deviation using the calculator
Enter x into L1
Enter the probabilities into L2
Press STAT
Arrow right to CALC
Select 1: 1-Var Stats L1,L2
Press ENTER
μ=
σ=
3) Identify usual and unusual outcomes using the range rule of thumb

Identifying Unusual Results with the Range Rule of Thumb
Maximum usual value =
Minimum usual value =
  2
  2
Let’s find the usual range for our experiment
[   2 ,
  2 ] =
Usual outcomes are all outcomes that are in that interval: ______________________
Unusual outcomes are all outcomes from the experiment which are outside of the interval:
___________________
6
Section 5.2 & 5.3 – Binomial Experiments

Features of a binomial experiment (5.2)
1)
2)
3)
4)
The experiment has a fixed number of trials (n)
The trials must be independent
Each trial has 2 possible outcomes: success (S) and failure (F)
Probabilities remain constant for each trial.
p is the probability of success, and q is the probability of failure
When sampling without replacement, the events can be treated as if they were independent
if the sample size is no more than 5% of the population size. (That is, n  0.05 N )

Find binomial probabilities with a shortcut feature of the calculator
To find individual probabilities: Use binompdf(n,p,x)
Press 2nd VARS
Select 0:binompdf(
Type n,p,x)
Press ENTER
To calculate cumulative probabilities from 0 to x, use binomcdf(n,p,x)

Mean, Variance, and Standard Deviation for the Binomial Distribution (5.3)
In Section 5.1, we used the general formulas for any discrete probability distribution. But the
special qualities of binomials distributions, lead to specialized formulas for binomials
  np
  npq
7
4) Experiment: Rolling a die 5 times and counting the number of 6s obtained
a) Is this a binomial experiment? Explain
b) What is n, what is p?
c) What are the possible values for the random variable x, the number of 6s obtained when we roll a die 5
times?
d) Let’s simulate this experiment:
Do RandInt(1,6,5) and record the number of 6s obtained in that roll
X (number of 6s obtained
when we roll the die 5 times)
Tally/Frequency
Experimental probability
Discuss all of the following:
- Histogram, shape of distribution (in relation to p)
- Probability of exactly x successes (from the table and from theory)
- Probability of at least x successes (from the table and from theory)
- Probability of at most x successes (from the table and from theory)
- Mean and standard deviation from the table
- Mean and standard deviation from theory (   np ,  
- Usual and unusual results with the probability rule
- Usual and unusual results with the range rule of thumb
npq )
8