2.10. Bivariate probabilities
Joint and marginal probabilities
Suppose
that a problem involves two distinct sets of events that we label
,
.
The events
and
are
mutually exclusive and collectively exhaustive within their sets, but
intersections
can occur between all events from the two sets. These intersections
can be considered as a basic outcome of a random experiment. Two sets
of events considered jointly in this way, are called bivariate,
and the probabilities are called bivariate
probabilities.
Table 2.1.
|
|
. . .
|
…………………………………………………………………… …………………………………………………………………. ………………………………………………………………….
|
Definition:
In
the table 2.1., the intersection probabilities
are called joint probabilities. Marginal probability is the
probability of a single event without consideration of any other
event. Marginal probability can be computed by summing the
corresponding row or column.
Example:
All the 420 employees of a company were asked if they smoke or not and whether they are university graduates or not. Based on this information, the following two-way classification table was prepared.
Table 2.2.
|
University Not a university graduate graduate |
Smoker Nonsmoker |
35 80 130 175 |
In table 2.2. each box that contains a number is called a cell. There are four cells in table 2.2. Each cell gives the frequency for two characteristics.
For example, 35 employees in this group possess two characteristics: They are university graduates and smoke. We can interpret the number in other cells the same way.
By adding the row of totals and the column of totals to table 2.2, we write table 2.3.
Table 2.3.
|
University Not a university graduate graduate |
Total |
Smoker Nonsmoker |
35 80 130 175 |
115 305 |
Total |
165 255 |
420 |
Suppose one employee is selected at random from these 420 employees. This employee may be classified either on the basis of smoker or non-smoker alone or on the basis of university graduate or not. If only one characteristic is considered at a time, the employee selected can be a smoker, non-smoker, a university graduate, or not a university graduate. The probability of each of these four characteristics or events is called marginal probabilities because they calculated by dividing the corresponding row margins (totals for rows) or column margins (totals for the columns) by the grand total. For table 2.3., the marginal probabilities are calculated as follows:
can
be interpreted as “The probability that randomly selected employee
is a smoker is 0.274”. Similarly
Now, suppose that one employee is selected at random from these 420 employees. Furthermore, assume that it is known that this (selected) employee is a smoker. In other words, the event that the employee selected is a smoker has already occurred. What is the probability that the employee selected is a university graduate?
This probability, P (university graduate/smoker), as we know, is called the conditional probability, and it is read as “the probability that the employee selected is a university graduate given that this employee is a smoker”. The required conditional probability is calculated as follows:
.
Example:
For the data of table 2.3. calculate the conditional probability that a randomly selected employee is a non-smoker given that this employee is not a university graduate.
Solution:
We are to compute the probability P(nonsmoker / not a university graduate).
.
The probability that randomly selected employee who is not a university graduate, does not smoke is 0.686.
Example:
Refer to the information on 420 employees given in table 2.3., are the events “smoker (S)” and “university graduate (U)” independent?
Solution:
If the occurrence of one event affects the probability of the occurrence of the other event then the two events are said to be dependent events. Using probability notation, the two events will be dependent if either
or
.
Events
S and U will be independent if
,
otherwise they will be dependent.
Using the information given in table 2.3., we compute the following two probabilities
;
and
.
Because these two probabilities are not equal, the two events are dependent. Here, dependence of events means that percentage of smokers is different from percentage between a university graduates.
Example:
A recent survey asked 100 people if they though women should be permitted to participate in weightlifting competition. The results of the survey are shown in the table.
Gender |
Yes |
No |
Total |
Male |
32 |
18 |
50 |
Female |
8 |
42 |
50 |
Total |
40 |
60 |
100 |
Find the probabilities
a) That a randomly selected person is a male.
b) The respondent answered “yes”, given that the respondent was a female
c) The respondent was a male, given that the respondent answered “no”.
Solution:
Let M=respondent was a male; Y= respondent answered “Yes”
F=respondent was a female; N=respondent answered “No”.
a) We need to compute the probability P (male). The probability that randomly selected respondent is a male is obtained by dividing total number of row labelled “Male” (50) by the total number of respondents (100).
b)
The problem is to find
.
The rule states
The
probability
is
the number of females who responded “yes” divided by the total
number of respondents
.
The
probability
is
the probability of selecting a female:
.
Then
c)
The problem is to find
Exercises
|
B
|
A
|
0.25 0.12 |
|
0.40 |
probabilities concerning two events A and B.
a) Determine the missing entries.
b) What is the probability that A occurs and B does not occur?
c) Find the probability that either A or B occurs.
d) Find the probability that one of these events occurs and other does not.
2. A woman’s clothing store owner buys from three companies:
A, B, and C. The purchases are shown below:
Product |
Company A |
Company B |
Company C |
Dresses |
24 |
18 |
12 |
Blouses |
13 |
36 |
15 |
If one item is selected at random, find the following probabilities:
a) It is purchased from company A or it is a dress.
b) It was purchased from company B or company C.
c) It is a blouse or was purchased from company A.
3. In a statistics class there are 18 local and 10 foreign students: 6 of the foreign students are females, and 12 of the local students are males. If a student is selected at random, what is the probability that
a) randomly selected student is a local or a female?
b) randomly selected student is foreign or a female student?
c) randomly selected student is a local or a foreign student?
4. Two thousand randomly selected adults were asked if they think they are financially better off than their parents. The following table gives the two-way classification of the responses based on the education levels of the persons included in the survey and whether they are financially better off, the same, or worse off than their parents.
|
Education level Less than Hight More than High school school high school |
Better off Same Worse off |
140 450 420 60 250 110 200 300 70 |
Suppose one adult is selected at random from these 2000 adults, find the probability that this adult is
a) financially better off his (her) parents or high school student;
b) more than high school student or financially worse off than his (her) parents;
c) financially better off his (her) parents or financially worse off his (her) parents;
d) financially better off than his (her) parents
e) financially better off than his (her) parents given that he (she) has less than high school education;
f) financially worse off than his (her) parents given that he (she) has hight school education
g) financially the same as his (her) parents given that he (she) has more than high school eduction.
h) Are the events “better off ” and ”hight school education” mutually exclusive? What about the events “less than high school” and “more than high school”? Why or why not?
i) Are the events “worse off “and “”more than high school” independent? Why or why not?
5. Eighty students in a university cafeteria were asked if they favoured a ban on smoking in the cafeteria. The results of the survey are shown in the table.
Class |
Favour |
Oppose |
No opinion |
Freshmen Sophomore |
15 23 |
27 5 |
8 2 |
If a student is selected at random, find these probabilities:
a) He or she opposes the ban, given that the student is a freshman.
b) Given that the student favours the ban, the student is a sophomore.
6. The following table gives a two-way classification of 200 randomly selected purchases made at department store.
|
Paid by cash/check Paid by credit card |
Male Female |
24 46 77 53 |
If one of these 200 purchases is selected at random, find the probability that it is
a) made by a female
b) paid by cash/check
c) paid by credit card given that the purchase is made by a male
d) made by a female given that it is paid by cash/check
e) made by a female and paid by a credit card
f) paid by cash/check or made by a male
g) Are the events “female” and “paid by credit card” independent? Are they mutually exclusive? Explain why or why not.
7. Three cable channels (6, 8, and 10) have quiz shows, comedies, and dramas. The table gives proportions in the nine joint classifications
Channels
Type of show |
Channel 6 Channel 8 Channel 10 |
Quiz show Comedy Drama |
0.21 0.11 0.06 0.08 0.21 0.08 0.01 0.07 0.17 |
a) What proportion of shows is quiz show?
b) What proportion of shows does Channel 6 have?
c) If randomly selected show is quiz show, what is the probability that it was shown on channel 6?
d) If the show was shown on channel 10, what is the probability that it was comedy?
e) What is the probability that randomly chosen show is drama, or shown on channel 8, or both?
8. A supermarket manager classified customers according to whether their visits to the store as frequent or infrequent and whether they often, sometimes, or never make a purchase. The accompanying table gives the proportions of people surveyed in each of six joint classifications:
Making purchase
Frequency of visit |
Often Sometimes Never |
Frequent Infrequent |
0.12 0.48 0.19 0.07 0.06 0.08 |
a) What is the probability that a customer is both a frequent shopper and often purchases?
b) What is the probability that a customer who never makes purchase visits the store frequently?
c) Are the events “Never makes a purchase” and “Visits the store frequently” independent?
d) What is the probability that a customer who infrequently visits the store often makes a purchase?
e) Are the events “Often makes a purchase” and “Visits the store infrequently” independent?
f) What is the probability that a customer frequently visits the store?
g) What is the probability that customer never makes a purchase in this store?
h) What is the probability that a customer either frequently visits the store or never makes a purchase, or both?
9. The accompanying table shows proportions of salespeople classified according to marital status and whether or not they own stocks.
-
Own stocks
Martial
status
Yes No
Married
Single
0.64 0.13
0.17 0.06
a) What is the probability that a randomly chosen salesperson was married?
b) What is the probability that a randomly chosen salesperson does not own stocks?
c) What is the probability that randomly chosen single salesperson does not own stocks?
d) What is the probability that a randomly chosen salesperson who owns stocks was married?
10. Forty-two percent of employees in a large corporation were in favour of a modified health care plan, and 22% of the corporation employees favoured a proposal to change the work schedule. Thirty- four percent of those favouring the health plan modification favoured the work schedule change.
a) What is the probability that a randomly selected employee is in favour of both modified health care plan and the changed work schedule?
b) What is the probability that a randomly chosen employee is in favour of at least one of these two changes?
c) What is the probability that a randomly selected employee favouring the work schedule change also favours the modified health plan?
Answers
1.
b)
;
c)
;
d)
;
2. a) 67/118; b) 81/118; c) 88/118; 3. a) 6/7; b) 4/7; c) 1; 4. a) 0.78; b) 0.55;
c) 0.79; d) 0.505; e) 0.350; f) 0.300; g) 0.183; h) “Better off” and “high school education” are not mutually exclusive, “Less than high school” and “ more than high school” are mutually exclusive events; i) “Worse off” and “more than high school “ are not independent events; 5. a) 0.54; b) 0.61; 6. a) 0.65; b) 0.51; c) 0.66; d) 0.76; e) 0.27; f) 0.74; g) no and no; 7. a) 0.38; b) 0.3;
c) 0.55; d) 0.26; e) 0.57; 8. a) 0.12; b) 0.704; c) No; d) 0.333; e) No; f) 0.79; g) 0.27; h) 0.87; 9. a) 0.77; b) 0.19; c) 0.2609; d) 0.7901; 10. a) 0.1428;
b) 0.4972; c) 0.6491.