Download Briefly, what is probability (include in this the 3 "approaches" to

1. Briefly, what is probability (include in this the 3 "approaches" to probability discussed in the
early part of chapter 5).
Classical approach to Probability
Suppose there are 𝑛 outcomes for a random experiment, which are equally likely, mutually
exclusive and exhaustive. If 𝑚 of the 𝑛 outcomes are favorable to an event 𝐴, then the
𝑚
probability of the event is defined as 𝑃(𝐴) = .
𝑛
Frequency approach to probability
Let the experiment be repeated 𝑛 times. Suppose the event 𝐴 occurs f times and does not
occur in 𝑛 − 𝑓 times. Then 𝑓 is called the frequency of the event 𝐴 in 𝑛 repetitions and 𝑓/𝑛 is called the
relative frequency. The limit of frequency ration as 𝑛 becomes larger and larger and tends to infinity is
defined as the probability of the event 𝐴.
Axiomatic approach to probability.
Consider the collection of all events. Then the function 𝑃 defined for every event is a probability
function if it satisfies the following three axioms.
Axiom 1 (Axiom of nonnegativity)
If A is any event, then P(A)≥0
Axiom 2(axiom of certainty)
Let S be the sample space. Then P(S)=1
Axiom 3EAxiom of additivity)
If A and b are mutually exclusive events, then P(A∪ 𝐵) =P(A)+P(B).
2. Please (1) work the following problems, (2) tell me what rule or principle you used to solve
them and (3) then look at others' answers to check yourself and that person -- let's see if we
can come to consensus on them!
1. Ninety students will graduate from Lima Shawnee High School this spring. Of the 90 students,
50 are planning to attend college. Two students are to be picked at random to carry flags at
graduation.
a. What is the probability both of the selected students plan to attend college?
Total number of ways in which 2 students can be selected from 90 students = (
4005
Number of ways in which 2 students can be selected from 50 students = (
50×49
50
)=
= 1225
1×2
2
Probability that both of the selected students are planning to attend college
=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠 1225 245
=
=
= 0.3059
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑎𝑠𝑒𝑠
4005 801
90×89
90
)=
=
1×2
2
b. What is the probability one of the two selected students plans to attend college?
Total number of ways in which 2 students can be selected from 90 students = (
4005
90×89
90
)=
=
1×2
2
Number of ways in which 1 student can be selected from the group of 50 students and 1
student from the remaining group of 40 students = 50 × 40 = 2000
Probability that one of the selected students plans to attend college
=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠 2000 400
=
=
= 0.4994
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑎𝑠𝑒𝑠
4005 801
2. A survey of undergraduate students in the School of Business at Northern University revealed
the following regarding the gender and majors of the students:
Major
Gender Accounting Management Finance total
Male 100 150 50 300
Female 100 50 50 200
total 200 200 100 500
Let us define the following events.
Let 𝐹 denote the event a randomly selected student is a female.
Let M denote the event a randomly selected student is a male.
Let A denote the event a randomly selected student is a accounting major
Let B denote the event a randomly selected student is a management major.
Let C denote the event a randomly selected student is a finance major.
a.
What is the probability of selecting a female student?
Total number of students = 500
Number of female students = 200
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠
200
2
Probability that a randomly selected student is a female =
=
= = 0.4
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑎𝑠𝑒𝑠
𝑃(𝐹) = 0.4
b.
What is the probability of selecting a finance or accounting major?
Total number of students = 500
Number of finance major students = 100
500
5
Probability that a randomly selected student is a finance major student
=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑎𝑠𝑒𝑠
=
100
500
1
= = 0.2
5
𝑃(𝐶) = 0.2
Number of accounting major students = 200
Probability that a randomly selected student is a accounting major student
=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑎𝑠𝑒𝑠
=
200
500
2
= = 0.4
5
𝑃(𝐴) = 0.4
Then the event that a randomly selected student is finance major or accounting major is 𝐴 ∪ 𝐶.
A student cannot be both accounting major and finance major simultaneously. So the events 𝐴 and C
are mutually exclusive. By the axiom of additivity,
𝑃(𝐴 ∪ 𝐶) = 𝑃(𝐴) + 𝑃(𝐶) =
100 200 300
+
=
= 0.6
500 500 500
The probability that a randomly selected student is finance major or accounting major = 0.6
c. What is the probability of selecting a female or an accounting major? Which rule of
addition did you apply?
The required probability is 𝑃(𝐹 ∪ 𝐴)
By the addition theorem of probability,
𝑃(𝐹 ∪ 𝐴) = 𝑃(𝐹) + 𝑃(𝐴) − 𝑃(𝐹 ∩ 𝐴)
𝑃(𝐹) =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑒𝑚𝑎𝑙𝑒 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 200
=
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠
500
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑐𝑐𝑜𝑢𝑛𝑡𝑖𝑛𝑔 𝑚𝑎𝑗𝑜𝑟 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 200
=
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠
500
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑒𝑚𝑎𝑙𝑒 𝑎𝑐𝑐𝑜𝑢𝑛𝑡𝑖𝑛𝑔 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 100
𝑃(𝐹 ∩ 𝐴) =
=
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠
500
𝑃(𝐴) =
Then
𝑃(𝐹 ∪ 𝐴) = 𝑃(𝐹) + 𝑃(𝐴) − 𝑃(𝐹 ∩ 𝐴) =
c.
200 200 100 300 3
+
−
=
= = 0.6
500 500 500 500 5
Are gender and major independent? Why?
Gender and major are independent if and only if all of the following conditions is satisfied.
If any one of the conditions is not satisfied, then gender and major are not independent.
𝑃(𝐹 ∩ 𝐴) = 𝑃(𝐹)𝑃(𝐴)
𝑃(𝐹 ∩ 𝐵) = 𝑃(𝐹)𝑃(𝐵)
𝑃(𝐹 ∩ 𝐶) = 𝑃(𝐹)𝑃(𝐶)
𝑃(𝑀 ∩ 𝐴) = 𝑃(𝑀)𝑃(𝐴)
𝑃(𝑀 ∩ 𝐵) = 𝑃(𝑀)𝑃(𝐵)
𝑃(𝑀 ∩ 𝐶) = 𝑃(𝑀)𝑃(𝐶)
A
B
C
Total
M
100
150
50
300
F
100
50
50
200
Total
200
200
100
500
From the above table
300 3
=
500 5
200 2
𝑃(𝐴) =
=
500 5
100 1
𝑃(𝑀 ∩ 𝐴) =
= = 0.20
500 5
3 2
6
𝑃(𝑀). 𝑃(𝐴) = . =
= 0.24
5 5 25
𝑃(𝑀) =
Clearly
𝑃(𝑀 ∩ 𝐴) ≠ 𝑃(𝑀)𝑃(𝐴)
Therefore the events 𝑀 and 𝐴 are not independent.
Gender and major are not independent.
d.
What is the probability of selecting an accounting major, given that the person selected is a
male?
The required probability is the conditional probability of the event 𝐴 given that the event 𝑀 has
occurred. By the definition of conditional probability,
𝑃(𝑀 ∩ 𝐴) 1/5 1
𝑃(𝐴/𝑀) =
=
=
𝑃(𝑀)
3/5 3
e.
Suppose two students are selected randomly to attend a lunch with the president of the
university. What is the probability that both of those selected are accounting majors?
500
Total number of ways in which 2 students can be selected from 500 students = (
)=
2
500×499
= 124,750
1×2
Number of ways in which 2 accounting major students can be selected from 200 accounting
200×199
200
major students = (
)=
= 1,990
1×2
2
Probability that the 2 students are accounting major
𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑎𝑠𝑒𝑠
1,990
199
=
=
=
= 0.015952
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑎𝑠𝑒𝑠
124,750 12,475
1. Reynolds Construction Company has agreed not to erect all "look-alike" homes in a new
subdivision. Five exterior designs are offered to potential home buyers. The builder has
standardized three interior plans that can be incorporated in any of the five exteriors. How
many different ways can the exterior and interior plans be offered to potential home buyers?
There are 5 exterior designs and there are 3 interior designs.
Number of ways in which 1 exterior design from 5 exterior designs 1 interior design fro 3 interior
design can be selected = 5 × 3 = 15
1. A puzzle in the newspaper presents a matching problem. The names of 10 in one U.S.
presidents are listed column, and their vice presidents are listed in random order in the second
column. The puzzle asks the reader to match each president with his vice president. If you
make the matches randomly, how many matches are possible? What is the probability all 10 of
your matches are correct?
The total number ways in which 10 vice presidents can be arranged against the 10 presidents is
= 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3628800
Of the 3628800 arrangements only is in the correct order.
1
Therefore, the required probability is
3628800
1. What is a continuous probability distribution (use the information from Applied Statistics in
Business and Economics, Chapters 7) as compared with a discrete probability distribution which
is discussed in chapter 6 (same text)?
Let 𝑋 be a random variable. If the variable 𝑋 assumes a finite number of values or a countably
infinite number of values, then it is known as a discrete random variable. The number of
students in a randomly selected school or the number of children in a randomly selected family
are examples of discrete random variables.
If the variable assumes a continuum of values it is called a continuous random variable. A
continuous variable assumes all values in a given interval. The height of randomly selected
student, the weight of a new born baby etc are examples of continuous random variables.
1. How does the Empirical Rule apply to continuous discrete probability distributions (Chapter 6)?
Does the Empirical Rule always apply to discrete probability distributions? Why or why not?
Does the Empirical Rule apply to continuous probability distributions (Chapter 7)? If so, how
and when?
If the distribution is normal 68% of the data lie within 1 standard deviation from the mean, 99% lie
within 2 standard deviations from the mean and 99.7% lie within 3 standard deviations. If the
distribution is approximately normal, then approximately 68% of the data lie within 1 standard
deviation from the mean, approximately 99% lie within 2 standard deviations from the mean and
approximately 99.7% lie within 3 standard deviations. This the empirical rule. Since normal
distribution is a continuous distribution, the empirical rule is applied to continuous distributions.
1. Use Chapter 7 (Applied Statistics in Business and Economics) to address the following 2
problems:
o The mean starting salary for college graduates in the spring of 2004 was $36,280.
Assume that the distribution of starting salaries follows the normal distribution with a
standard deviation of $3,300. What percent of the graduates have starting salaries:
1. Between $35,000 and $40,000?
2. More than $45,000?
3. Between $40,000 and $45,000?
Let 𝑋 denote the starting salary of a randomly selected college graduate. Then X follows normal
𝑋−36,280
distribution with mean 36,280 and standard deviation 3,300. Then Z =
follows standard normal
3,300
distribution.
1. 𝑃(35,000 < 𝑋 < 40,000) = 𝑃 (
35,000−36,280
3,300
<
𝑋−36,280
3,300
<
40,000−36,280
3,300
) = 𝑃(−0.3879 < 𝑍 < 1.1273)
= 𝑃(𝑍 < 1.1273) − 𝑃(𝑍 < −0.3879) = 0.8702 − 0.3491 = 0.5211
2. 𝑃(𝑋 > 45,000) = 𝑃 (
𝑋−36,280
3,300
>
3. 𝑃(40,000 < 𝑋 < 45,000) = 𝑃 (
45,000−36,280
3,300
40,000−36,280
3,300
) = 𝑃(𝑍 > 2.6424) = 1 − 𝑃(𝑍, 2.6424) = 1 − 0.9959 = 0.0041
<
𝑋−36,280
3,300
<
45,000−36,280
3,300
) = 𝑃(1.1273 < 𝑍 < 2.6424)
= 𝑃(𝑍 < 2.6424) − 𝑃(𝑍 < 1.1273) = 0.9959 − 0.8702 = 0.1257
The price of shares of Bank of Florida at the end of trading each day for the last year
followed the normal distribution. Assume there were 240 trading days in the year. The
mean price was $42.00 per share and the standard deviation was $2.25 per share.
1. What percent of the days was the price over $45.00? How many days would you
estimate?
2. What percent of the days was the price between $38.00 and $40.00?
3. What was the stock's price on the highest 15 percent of days?
Let 𝑋 denote the price of shares of Bank of Florida on at the end of a randomly selected
trading day. Then 𝑋 follows normal distribution with mean 42.00 and standard deviation 2.25.
o
1. 𝑃(𝑋 > 45.00) = 𝑃 (𝑍 >
45.00−42.00
2.25
) = 𝑃(𝑍 > 1.3333) = 1 − 𝑃(𝑍 < 1.3333) = 1 − 0.9088 = 0.0912
The price was above 45.00 in 9.12% of days which is 240 ×
38−42
9.12
100
= 21.89 ≅ 22 𝑑𝑎𝑦𝑠
40−42
2. 𝑃(38 < 𝑋 < 40) = 𝑃 (
<𝑍<
) = 𝑃(−1.7778 < 𝑍 < −0.8889)
2.25
2.25
= 𝑃(𝑍 < −0.8889) − 𝑃(𝑍 < −1.7778) = 0.1870 − 0.0377 = 0.1493
The price was between 38 and 40 in 14.93% of the days.
3. From tables of standard normal distribution 𝑃(𝑍 > 1.0364) = 0.15 = 15%
The corresponding value of 𝑋 is 𝑋 = 42.00 + 1.0364 × 2.25 = 44.3320 ≅ 44.33
The stock’s price on the highest 15 percent of days was above $44.33
***********************************************************************
Statistical Symbols and Definitions Matching Assignment
Match the letter of the definition on the right to the appropriate symbol on the left.
Symbols
 1.
 2.
 3.
 4.
 5.
 6.
 7.
 8.
S (Uppercase Sigma) ____
m (Mu) ____
s (Lowercase Sigma) ____
p (Pi) ____
e (Epsilon) ____
c2 (Chi Square) ____
! ____
H0 ____
Definitions
 a. Null hypothesis
 b. Summation
 c. Factorial
 d. Nonparametric hypothesis test
 e. Population standard deviation
 f. Alternate hypothesis
 g. Maximum allowable error
 h. Population mean

9. H1 ____
i. Probability of success in a binomial trial
Symbol
Definition
Σ
Summation
𝜇
Population mean
𝜎
Population standard deviation
𝜋
Probability of success in binomial trial
𝜖
Maximum allowable error
𝜒2
Nonparametric hypothesis test
!
Factorial
𝐻0
Null hypothesis
𝐻1
Alternate hypothesis
Match the letter of the term on the right to the definition of that term on the left.
Definitions
Terms
 1. The average of the squared deviation scores from a
 a. Reliability
distribution mean. ____
 2. Midpoint in the distribution of numbers. ____
 b. Mode
 3. It has to do with the accuracy and precision of a
 c. Generalization
measurement procedure. ____
 4. Examines if an observed causal relationship
 d. Variance
generalizes across persons, settings, and times. ____
 5. The difference between the largest and smallest
 e. Median
score in a distribution. ____
 6. The arithmetic average. ____
 f. External validity
 7. Refers to the extent to which a test measures what
 g. Mean
we actually wish to measure. ____
 8. The most frequently occurring value in a set of
 h. Internal validity
numbers. ____
 9. The conclusion from research conducted on a sample  i. Range
population to the population as a whole. ____
 j. Standard deviation
 10. Examines whether the conclusion that we draw
 k. Validity
about a demonstrated experimental relationship truly
implies cause. ____
 11. Determines how far away the data values are from
the average. ____
Definitions
The average of the squared deviation scores from
a distribution mean. ____
. Midpoint in the distribution of numbers. ____
It has to do with the accuracy and precision of a
Terms
Variance
Median
Reliability
measurement procedure. ____
. Examines if an observed causal relationship
generalizes across persons, settings, and times.
____
The difference between the largest and smallest
score in a distribution. ____
The arithmetic average. ____
. Refers to the extent to which a test measures
what we actually wish to measure. ____
The most frequently occurring value in a set of
numbers. ____
The conclusion from research conducted on a
sample population to the population as a whole.
____
Examines whether the conclusion that we draw
about a demonstrated experimental relationship
truly implies cause. ____
Determines how far away the data values are
from the average. ____
External validity
Range
Mean
Validity
Mode
Generalization
Internal validity
Standard deviation