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
Total	165 255

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
Female
Total

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

Дата добавления: 2015-08-05